Generalized Convexity: A contemporary vision about Convexity

Generalized Convexity: A contemporary vision about Convexity Miguel J. Vivas C.1 Jorge E. Hernández H.2 1 Ponti…cia Universidad Católica del Ecuador Facultad de Ciencias Exactas y Naturales, Escuela de Ciencias Físicas y Matemática. Sede Quito, Ecuador. Universidad Centroccidental Lisandro Alvarado Decanato de Ciencias Económicas y Empresariales Barquisimeto, Venezuela. 2 November, 2017 ii Contents Introduction xi 1 Convex Function on the Real Line 1.1 Introduction . . . . . . . . . . . . . . 1.2 Continuity and Di¤erentiability . . . 1.3 Characterizations . . . . . . . . . . . 1.4 Closure under Functional Operations. 2 Inequalities. 2.1 Introduction . . . . . . . . . . . 2.2 Classical inequalities . . . . . . 2.3 Jensen’s Inequality . . . . . . . 2.4 Hermite-Hadamard’s Inequality 2.5 Fejér-Hadamard inequality . . . 2.6 Ostrowski’s Inequality . . . . . 2.7 Chebysev ’s inequality . . . . . 2.8 Grüss’s Inequality . . . . . . . . 2.9 Simpson’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 5 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 11 12 13 13 14 14 14 15 15 3 Generalized Convexity 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . 3.2 P Convex Functions . . . . . . . . . . . . . . . . 3.3 s Convex Functions in the …rst and second sense 3.4 m Convex Functions . . . . . . . . . . . . . . . . 3.5 h Convex Functions . . . . . . . . . . . . . . . . 3.6 Convex Functions . . . . . . . . . . . . . . . . 3.7 GA Convex Functions . . . . . . . . . . . . . . . 3.8 (s; m) Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 17 17 17 18 19 20 20 21 iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv CONTENTS 3.9 3.10 3.11 3.12 (m; h1 ; h2 ) Convex function . . . . . . . . . . . . (m; h1 ; h2 ) GA Convex function . . . . . . . . Relative m-logarithmically- semi-convex function Other de…nitions regarding generalized convexity 3.12.1 Quasi convex Functions. . . . . . . . . . . 3.12.2 Wright convex functions . . . . . . . . . . 3.12.3 Strongly convex functions . . . . . . . . . 3.12.4 Strongly (s; m) convex functions . . . . . 3.12.5 Relative Strongly h Convex Functions . . 3.12.6 Invex Convex Functions . . . . . . . . . . 3.12.7 convex function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Inequalities and Generalized Convexity 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 4.2 For P Convex Functions . . . . . . . . . . . . . . . . 4.3 For s Convex Functions . . . . . . . . . . . . . . . . 4.4 For h Convex Functions . . . . . . . . . . . . . . . . 4.4.1 In the Self adjoint operators in Hilbert Spaces 4.4.2 In the Real Line in Fractal Sets Environment 4.5 For m Convex Functions . . . . . . . . . . . . . . . 4.6 For GA Convex Functions . . . . . . . . . . . . . . . 4.7 For (s; m) Convex Functions . . . . . . . . . . . . . 4.8 For strongly (s; m) Convex Functions . . . . . . . . 4.9 For Relative Strongly h Convex Functions . . . . . . 4.10 For (m; h1 ; h2 ) Convex Functions . . . . . . . . . . . 4.11 For (m; h1 ; h2 ) GA Convex Functions . . . . . . . 5 Open Problems 5.1 Problem #1 . 5.2 Problem #2 . 5.3 Problem #3 . 5.4 Problem #4. . 5.5 Problem #5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 22 22 24 24 24 25 25 25 26 27 . . . . . . . . . . . . . 29 29 29 30 31 31 53 56 58 60 63 67 70 72 . . . . . 77 77 79 80 80 81 A About Self-adjoint Operators in Hilbert Spaces 83 B About Convex Stochastic Processes 87 CONTENTS v C About Real Line on Fractal Sets 91 Afterword 95 vi CONTENTS Preface In view of the increasing information obtained by the current development and evolution of the concept of convexity within the framework of Functional Theory we have proposed to put something of our part in the elaboration of this small summary and development of the mentioned aspect, taking advantage of the occasion of the XIII International Meeting of Mathematics (Ecuentro Internacional de Matemáticas) sponsored by the University of the Atlantic (Colombia). In addition to such a proposal, this minibook will support the course given at this seminar. We believe that this contribution can promote the origin and development of a group of researchers in the mentioned area in such a way that, at an international level, as is the framework of these conferences, this line of research has a greater growth in knowledge and academic development for consolidated researchers and beginning researchers. Vivas Cortez, Miguel José (PUCE) Hernández Hernández, Jorge Eliécer (UCLA) vii viii PREFACE About the Authors Miguel Vivas Cortez, was born in Acarigua, Portuguesa State (Venezuela) in the year of 1972, is graduated as Mathematician of the Centroccidental University Lisandro Alvarado (UCLA), where he also obtains his master’s degree in mathematics, pure mathematical mention, his thesis was published in the prestigious journal (Q1 Scopus) Nonlinear Analysis: Real World Applications (United Kingdom). He received his PhD in Sciences, Mathematical Mention from the Central University of Venezuela, his doctoral thesis generated 4 publications in prestigious international journals. He has more than 45 scienti…c publications, has more than one hundred citations, has directed several graduate thesis , in lines of research such as: Di¤erential Equations (Ecological Models), Nonlinear Analysis (Generalized Convexity) and Functional Analysis (Generalized Bounded Variation Functions), is the author of several textbooks among which stand out: Analysis in a Complex Variable ( 2009), Di¤erential Calculus for Science and Technology (2012) and Linear Algebra, a Practical Approach (2013) and Mathematics Prior to Calculus (2014), among others. Dr. Vivas Cortez has more than 20 years of experience as a teacher in mathematics in undergraduate and postgraduate university education, he entered by competitive examination in complex analysis at the Universidad Centroccidental Lisandro Alvarado (UCLA) in 1997, where he worked until October 2017, he has been a visiting professor at the National Arturo Prat University (Chile), a titular professor at UCLA and was a visiting professor and an occasional professor at the Escuela Politécnica del Litoral (ESPOL), Guayaquil, Ecuador in 2016. He also develops activity of dissemination in scienti…c subjects related to mathematics through informative and scienti…c conferences in various institutions. He has been the winner in Venezuela of the research stimulus award (PEI) and the research promotion award (PPI). He is currently Principal Professor 3, at the Ponti…cia Universidad Católica de Ecuador (PUCE), Quito Headquarters. PREFACE ix Jorge E. Hernández H, was born in Barquisimeto, Lara State (Venezuela) in the year of 1956. He graduated with a degree in Mathematical Sciences from the Centroccidental University Lisandro Alvarado (UCLA), where he also obtained his Master’s Degree in Mathematics, Mathematics Pure, his thesis corresponds to a Characterization of the BMO Spaces using Wavelets through the Spaces of Triebel-Lizorkin. He is Associate Professor at the Universidad Centroccidental Lisandro Alvarado in the Deanship of Economic and Business Sciences attached to the Department of Quantitative Techniques at the University. The areas of Mathematics in which it has developed are: Analysis, Numerical Analysis, Di¤erence and Integral Calculus, Harmonic Analysis, Statistics, Probability and Operations Research. He has also been a juror of numerous Ascent, degree and internship jobs, as well as a permanent member of the Lisandro Alvarado Prize Committee. Currently, in the area of research is in charge of the research line corresponding to Generalized Convexity, as well as topics in Harmonic Analysis. x PREFACE Introduction We want to start this mini book with a thought of Bertrand Russell (18721970): "Mathematics possesses not only the truth, but a certain supreme beauty. A cold and austere beauty, like that of a sculpture". With these words we want to emphasize the usefulness of what is presented in this paper for the development of a knowledge structure that is already being developed internationally and that requires our contribution. This minibook is structured in 4 chapters. Chapter one develops the concept of convexity and its basic properties: de…nition, relations with continuity and di¤erentiability of this concept with these properties, algebraic closure with respect to some basic operations. Chapter 2 gives us a mathematical environment, known as Inequalities, classical in Mathematics and from which have produced important results; we mention in this paper: some classical inequalities in numerical theory, Jensen’s inequality, Hermite-Hadamard inequality, and Ostrowski’s inequality. In chapter 3 we show and develop some basic properties of the concepts of generalized convexity obtained in the last years: m-convexity, s-convexity, h-convexity and GA-convexity. In Chapter 4 we establish and demonstrate the results obtained in recent years about the generalized convexity in the environment of inequalities, including their relations with, for example, self-adjoint operators in Hilbert Spaces and Stochastic Processes. We also provide 3 Appendices that partially cover the basic knowledge required in the areas of self-adjoint operators in Hilbert Spaces, Stochastic Processes and the Real Straight in fractal sets. xi xii INTRODUCTION Chapter 1 Convex Function on the Real Line 1.1 Introduction As Nicolescu C. and Persson, L. wrote in [86]: "Convexity is a simple and natural notion which can be traced back to Archi- medes (Circa 250 B.C) in connection with his famous estimate of the value (using inscribed and circumscribed regular polygons). He noticed the importatnt fact that the perimeter of a convex …gure is smaller than the perimeter of any other convex …gure, surrounding it." In the above reference quotation we are located in very ancient times where the concept of "convexity" literally did not exist but the geometric entity itself. Jensen’s work (See [61]) identi…ed and named it, and wrote: "It seems to me that the notion of convex function is just a fundamental as positive function or increasing function. If I am not mistaken in this, the notion ought to …nd it place in elementary expositions of the theory of real functions" Nowadays, we manipulate his de…nition. De…nition 1 A function f : I ! R is called convex if the inequality f ( x + (1 )y) f (x) + (1 1 )f (y) (1.1) 2 CHAPTER 1. CONVEX FUNCTION ON THE REAL LINE holds for all x; y 2 I and 2 [0; 1] : It is called strictly convex if the inequality (1.1) holds strictly whenever x and y are distinct points and 2 (0; 1) : If f is convex (strictly convex) then we say that f is concave (respectively, strictly concave). If f is both, convex and concave, then f is said to be a¢ne. Particulary, if = 1=2 we have the well known midpoint convex function or Jensen’s inequality f x+y 2 f (x) + f (y) 2 for all x; y 2 I: In some publications you can …nd a characterization of convexity in these terms: a function f : I ! R is convex if and only if f (x + t(y x)) f (x) + t(f (y) f (x)) (1.2) for all x; y 2 I and t 2 [0; 1] : The convexity of functions plays a signi…cant role in many …elds, for example, in biological system, economy, optimization and so on [49, 102]. And many important inequalities are established for the class of convex functions. Convexity is one of the hypotheses often used in optimization theory. It is generally used to give global validity for certain propositions, which otherwise would only be true locally. Some properties of these kind of functions can be found in the book of Nicolescu C. and Peerson L. [86] , and the book of Roberts and Varberg [100]. 1.2 Continuity and Di¤erentiability Let’s start with the following property: A convex function f is bounded from above on a closed interval [a; b] by M = max (f (a); f (b)) : It is following from the next fact: if z 2 [a; b] then z = a + (1 )b for some 2 [0; 1], so, using the convexity of f , we have f (z) = f ( a + (1 M + (1 )b) f (a) + (1 )M = M: )f (b) 3 1.2. CONTINUITY AND DIFFERENTIABILITY Also, f is bounded from below, in fact , if we write an arbitrary point z 2 [a; b] as z = (a + b) =2 + t, we have f a+b t t a+b t a+b + =f + + 4 2 2 4 2 4 1 1 a+b a+b f +t + f t 2 2 2 2 a+b 2 = f 2 t 2 and from here f a+b +t 2 a+b 2 2f f a+b 2 t : Using the fact that f is bounden from above by M we get a+b +t 2 f 2f a+b 2 M = m: A convex function may be noncontinuous at the ends of its domain; may have jump discontinuities. In the interior is not only continuous but enjoys a stronger condition. Theorem 2 If f : I ! R is a convex function then f satisfy a Lipchtiz condition on any closed interval [a; b] int(I): Consequently, f is absolutely continuous on [a; b] and continuous in int(I): Proof. Let " > 0 arbitrary so that a "; b + " 2 I: Let m and M the lower and upper bounds for f in [a "; b + "] : If x; y are distinct points in [a; b] ; set " jy xj z=y+ (y x) ; = : jy xj " + jy xj Then z 2 [a "; b + "] ; y = z + (1 f (y) f (z) + (1 )x, and we have )f (x) = (f (z) f (x)) + f (x) from here f (y) f (x) jy xj (f (z) " + jy xj jy xj (M m) " = K jy xj f (x)) 4 CHAPTER 1. CONVEX FUNCTION ON THE REAL LINE where K = (M m)=": Since this is true for all x; y 2 [a; b] we conclude that jf (y) f (x)j K jy xj as desired. Recall that a function f is absolutely continuous over an interval [a; b] if for any " > 0 there exists > 0 such that for any collection f(ai ; bi )gni=1 of disjoint intervals of [a; b] with ni=1 (bi ai ) < we have n X i=1 jf (bi ) f (ai )j < ": In our case, the choice = "=K …ll the requirement. The continuity of f in int(I) follows from the arbitrariness of [a; b] : About the di¤erentiability we have the following result. Proposition 3 Let f : I ! R be a convex function , x; y; z 2 I such that x < y < z then f (y) y f (x) x f (z) z f (x) x f (z) z f (y) : y Proof. For the …rst inequality , let’s do y =x+ y z x (z x x) and use the inequality (1.2). For the second inequality we do z =x+ z z x (z y y) and and we proceed as before. Corollary 4 Let f : I ! R be a convex function, then for all x 2 I, the function f (t) f (x) t 2 I fxg ! t x is increasing. 5 1.3. CHARACTERIZATIONS Using Proposition 3 and Corollary 4 is easy to prove that f : I ! R has f and f+0 . This result was proved by Otto Stolz in [112]. 0 Theorem 5 Let f : I ! R a function convex, then f has lateral derivatives at each point of I, the lateral derivatives are increasing and the set E of the points of I, where f is not derivable is countable, and f is continuous in I E: We left to the reader the proof. 1.3 Characterizations Often, the mathematicians recognize convex functions by properties of the derivatives, by an integral representation, or by geometric properties of the graph. Theorem 6 f : [a; b] ! R is convex (strictly convex) if and only if there is an increasing (strictly increasing) function g : [a; b] ! R and a point c 2 (a; b) such for all x 2 (a; b) we have Z x g(t)dt: (1.3) f (x) f (c) = c Proof. First, suppose that f : [a; b] ! R is convex function. Choose g = f 0 ; which, from Theorem 5, exists and is increasing. Let c 2 (a; b) any point. By Theorem 2, f is absolutely continuous in [c; x] : By a classical Theorem for Lebesgue integral (see Royden H.L. [101]), we have Z x g(t)dt: f (x) f (c) = c Moreover, if f is strictly convex then g is strictly convex. Conversely, suppose that (1.3) holds with g increasing. Let 2 [0; 1] then for x < y in [a; b] we have Z x+(1 )y Z y g(t)dt: g(t)dt f (x)+(1 )f (y) f ( x+(1 )y) = (1 ) x+(1 )y x Replacing the integrand in the right side of the previous equality bye the constant g ( x + (1 )y), we obtain (1 )g ( x + (1 )y) [y x (1 )y] g ( x + (1 )y) [ x + (1 )y x] 6 = (1 CHAPTER 1. CONVEX FUNCTION ON THE REAL LINE )g ( x + (1 )y) [y x] g ( x + (1 )y) ( 1) [y x] = 0: So f (x) + (1 )f (y) f ( x + (1 )y) 0; therefore f is a convex function. Theorem 7 Suppose f : [a; b] ! R is di¤erentiable on (a; b) : Then f is a convex function if and only if f 0 is increasing. Proof. Suppose that f 0 is increasing. Then the fundamente theorem of calculus ensure us that Z c f 0 (t)dt f (x) f (c) = x for any c 2 (a; b) : Therefore, using Theorem 6, we can conclude that f is a convex function. Theorem 8 Suppose f 00 exists on (a; b) : Then f is a convex function if and only if f 00 (x) 0: And if f 00 (x) > 0; then f is strictly convex on the interval. Proof. Under the given assumption,f 0 is increasing if and only if f 00 is nonnegative and f 0 is strictly increasing when f 00 is positive. This combined with Theorem 7 gives us our result. For the next characterization theorem we will need the following de…nition. De…nition 9 We say that a function f de…ned on an interval I has a support in x0 2 I if there exists an a¢ne function A(x) = f (x0 ) + m(x x0 ) such that A(x) f (x) for every x 2 I: The graph of the a¢ne function A is called a line of support for f in x0 : Theorem 10 f : (a; b) ! R is a convex function if and only if there is at lest one line of support f at each x0 2 (a; b) : Proof. If f is a convex function and x0 2 (a; b) ; choose m 2 f 0 (x0 ) ; f+0 (x0 ) , then f (x) f (x0 ) f (x) f (x0 ) m or m x x0 x x0 7 1.4. CLOSURE UNDER FUNCTIONAL OPERATIONS. according x > x0 or x < x0 , respectively. In either case, f (x) m (x x0 ), that is f (x) m (x f (x0 ) x0 ) + f (x0 ) = A(x): Conversely, suppose that f has a support line at each x0 2 (a; b) : Let x; y 2 (a; b) : If x0 = x + (1 )y; 2 [0; 1] ;let A(x) = f (x0 ) + m(x x0 ) a support line for f at x0 : Then f (x0 ) = A(x0 ) = A ( x + (1 )y) = A(x)+(1 )A(y) f (x)+(1 )f (y) as desired. 1.4 Closure under Functional Operations. In this section we will establish some theorems that guarantee the convexity of a function, which is the result of the sum of convex functions, multiplication by scalar, composition of convex functions, supreme of a family of convex functions and limit of a succession of convex functions. Theorem 11 If f : I ! R and g : I ! R are convex functions and then (f + g) and f are convex functions. 0; Proof. The proof follows from the de…nition of convex function (1.1). Theorem 12 Let f : I ! R and g : J ! R where range(f ) J: If f and g are convex functions and g is increasing then (g f ) is a convex function on I: Proof. The Theorem follows from g (f ( x + (1 )y)) g ( f (x) + (1 g(f (x)) + (1 (g f ) ( x + (1 )y) (g f ) (x) + (1 )f (y)) )g(f (y)); that is The proof is complete. ) (g f ) (y): 8 CHAPTER 1. CONVEX FUNCTION ON THE REAL LINE Theorem 13 If f : I ! R and g : I ! R are both non-negative, decreasing (increasing) and convex functions, then h(x) = f (x)g(x) is non-negative, decreasing (increasing) and convex function. Proof. The …rst two properties are easy to check. We will proof the convexity. Note that for x < y we have [f (x) f (y)] [g(y) g(x)] 0 which implies that f (x)g(y) + f (y)g(x) Now if 2 (0; 1) we have f ( x + (1 )y) g ( x + (1 [ f (x) + (1 = f (x)g(x) + f (y)g(y) )y) )f (y)] [ g(x) + (1 )g(y)] 2 f (x)g(x) + (1 ) [f (x)g(y) + f (y)g(y)] + (1 )2 f (y)g(y) 2 f (x)g(x) + (1 ) [f (x)g(x) + f (y)g(y)] + (1 )2 f (y)g(y) = f (x)g(x) + (1 )f (y)g(y): The proof is complete. It is not di¢cult to prove the following theorems, consequently they are left to the reader. Theorem 14 Let f : I ! R be an arbitrary family of convex functions and f = sup (f ): If J = fx 2 I : f (x) < 1g then J is an interval and f is convex on J: Theorem 15 If ffn g1 n=1 is a sequence such that fn : I ! R is a convex function for all n = 1; 2; 3; ::: , converging to a …nite limit function f on I; then f is a convex function. Moreover the convergence is uniform on any closed subinterval of int(I): De…nition 16 We shall say that f is log-convex on an interval I if f is positive and log(f ) is convex on I: This is equivalent to requiring that f be positive and satisfy f ( x + (1 for x; y 2 I and 2 (0; 1) : )y) f (x)f 1 (y) 1.4. CLOSURE UNDER FUNCTIONAL OPERATIONS. 9 The class of log-convex functions have some closure properties which are exposed in the following theorem and are proposed to the reader for veri…cation. Theorem 17 The class of log-convex functions on an interval I is closed under the addition, multiplication, and taking of limits, provided that the limit exists and is positive. 10 CHAPTER 1. CONVEX FUNCTION ON THE REAL LINE Chapter 2 Inequalities. 2.1 Introduction Inequalities are one of the most important instrument in many branches of Mathematics such as Functional Analysis, Theory of Di¤erential and Integral Equations, Probability Theory, etc. They are also useful in mechanics, physics and other sciences. A systematic study of inequalities was started in the classical book of G.H. Hardy, J.E. Littlewood and G. Pólya [53] and continued by E.F. Beckenbach and R. Bellman in [11]. As Beckenbach and Bellman wrote in his book [11] "... an enormous account of e¤orts has been devoted to the sharpening and extension of the classical inequalities, to the discovery of new types of inequalities, and to the applications of inequalities in many parts of Analysis." Nowadays the theory of inequalities is still being intensively developed. This fact is con…rmed by a great number of recent published books [10, 135] and a huge number of articles on inequalities [4, 5, 6, 18, 21, 22, 37, 45, 71, 104, 113, 130]. Thus, the theory of inequalities may be regarded as an independent area of mathematics. The Ostrowski’s inequality was introduced by Alexander Ostrowski in [94], and with the passing of the years, generalizations on the same, involving derivatives of the function under study, have taken place. 11 12 CHAPTER 2. INEQUALITIES. 2.2 Classical inequalities In this section we will brie‡y present some classical inequalities. The following inequality is known as Arithmetic-Geometric. Theorem 18 For any non-negative numbers a and b; we have p a+b 2 ab: The equality holds if and only if a = b: It is possible generalize this last inequality for n numbers. Theorem 19 (See [11], Theorem 1)For any non-negative numbers a1 ; a2 ; :::; an we have v u n Pn uY i=1 ai t ai n i=1 The equality holds if and only if a1 = a2 = ::: = an : This other is called the Bernoulli’s inequality. Theorem 20 (See [80]). If h < Bernoulli’s inequality states that 1 and n is a natural number , then (1 + h)n 1 + nh: Also, there is another famous inequality called as Chebychev’s inequality. Theorem 21 (See [80])If a1 Chebychev’s inequality state ! n 1X ai n i=1 a2 ::: n 1X bi n i=1 ! an and b1 b2 ::: bn , then n 1X ai b i : n i=1 For the demonstration of any of these classic inequalities we refer the reader to Mitrinovic’s book [79] and Beckenbach and Bellman’s book [11]. 13 2.3. JENSEN’S INEQUALITY 2.3 Jensen’s Inequality Jensen’s inequality is sometimes called the king of inequalities since it implies the whole series of other classical inequalities (e.g. those by Hölder, Minkowski, Beckenbach-Dresher and Young, the arithmetic-geometric mean inequality etc.). Jensen’s inequality for convex functions is probably one of the most important inequalities which is extensively used in almost all areas of mathematics, especially in mathematical analysis and statistics. For a comprehensive inspection of the classical and recent results related to the inequality (1.1) the reader is referred to [81, 97, 109]. The classical Jensen’s inequality is contained in the following theorem. Theorem 22 (See [61]) Let f : I R ! R be a P convex function over I. Then for every xi 2 I; ti 2 [0; 1] , i = 1; 2; ::; n, and ni=1 ti = 1 , we have ! n n X X ti xi f ti f (xi ): (2.1) i=1 2.4 i=1 Hermite-Hadamard’s Inequality On November 22, 1881, Hermite (1822-1901) sent a letter to the journal Mathesis. An extract from that letter was published in Mathesis 3 (1883, p. 82). It is well-known that one of the most fundamental and interesting inequalities for classical convex functions is that associated with the name of Hermite-Hadamard’s inequality which provides a lower and an upper estimations for the integral average of any convex functions de…ned on a compact interval, involving the midpoint and the endpoints of the domain. Historically, the statement for that journal was as follows. Theorem 23 (See [50]) Let f be a convex function over [a; b], a < b. If f is integrable over [a; b], then Z b a+b f (a) + f (b) 1 f f (x) dx : (2.2) 2 (b a) a 2 The above inequality ( 2.2 )was …rstly discovered by Hermite in 1881 in the journal Mathesis (see Mitrinović and La¼cković [80]). But, this beautiful result was nowhere mentioned in the mathematical literature and was not 14 CHAPTER 2. INEQUALITIES. widely known as Hermite’s result (see Klariµcić et al. [66]). For more recent results which generalize, improve, and extend the classical Hermite-Hadamard inequality (2.2), see for instance [65, 96, 97], and references therein. HermiteHadamard’s inequality has several applications in nonlinear analysis and the geometry of Banach spaces, see [64, 88]. 2.5 Fejér-Hadamard inequality In the year 1905 Leopold Féjer [42] gave a generalization of the inequality (2.2) as follows: Theorem 24 If f : [a; b] R ! R is a convex function, and g : [a; b] ! R a+b is nonnegative, integrable and symmetric about , then 2 Z b Z b Z f (a) + f (b) b a+b f f (x)g(x)dx g(x)dx: (2.3) g(x)dx 2 2 a a a 2.6 Ostrowski’s Inequality Ostrowski’s inequality was introduced by Alexander Ostrowski in [94], and with the passing of the years, generalizations on the same, involving derivatives of the function under study, have taken place. Ostrowski’s Inequality. Let f : I [0; +1) ! R be a di¤erentiable 0 function on int(I), such that f 2 L[a; b], where a; b 2 I with a < b. If jf 0 (x)j M , then the inequality: Z b (x a)2 + (b x)2 M 1 (2.4) f (x) dx f (x) (b a) a b a 2 holds. 2.7 Chebysev ’s inequality µ In 1882, P. L. Cebyšev (See [82]) proved that, if f 0 ; g 0 2 L1 [a; b]; then, Z b Z b Z b 1 1 1 f (x)g(x)dx f (x)dx g(x)dx b a a b a a b a a 15 2.8. GRÜSS’S INEQUALITY 1 (b 12 2.8 a)2 kf 0 k1 kg 0 k1 : Grüss’s Inequality In 1934, G. Grüss (See [82]) showed that 1 b a Z b a f (x)g(x)dx 1 b a Z b 1 f (x)dx b a a Z b g(x)dx a 1 (M m)(N 4 provided M; m; N; n are real numbers satisfying the conditions, for all x 2 [a; b] 2.9 1<m f (x) M 1 1<n g(x) N 1 n) Simpson’s Inequality The following inequality, named Simpson’s inequality, is one of the best known results in the mathematical literature Theorem 25 (See [105])Let f : [a; b] ! R be a four times continuously di¤erentiable function on (a; b) and f (4) 1 = supx2(a;b) f (4) (x) < 1: Then the following inequality holds Z b 1 1 a+b 1 f (a) + f (b) f (x) dx f (4) 1 (b a)4 + 2f 3 2 2 b a a 2880 16 CHAPTER 2. INEQUALITIES. Chapter 3 Generalized Convexity 3.1 Introduction In recent years several extensions and generalizations have been considered for classical convexity, and the theory of inequalities has made essential contributions to many areas of Mathematics. The concept of convexity has been generalized depending on the problem and applications studied. Some of these generalizations are quasi convex, h-convex, Wright-convex , strongly convex , P convex, s convex in the …rst sense and second sense, strongly convex, m convex, (s; m) convex, (m; h1 ; h2 ) convex , invex, between others. 3.2 P Convex Functions De…nition 26 [31]We say that f : I ! R is a P-function, or that f belongs to the class P (I), if f is a non-negative function and for all x; y 2 I; t 2 [0; 1] we have f (tx + (1 t)y) f (x) + f (y): 3.3 s Convex Functions in the …rst and second sense In 1994, H. Hudzik and L. Maligranda [58] set out a set of properties of s convex functions de…ned in the …rst instance by W. Orlicz [93] in 1961 17 18 CHAPTER 3. GENERALIZED CONVEXITY and in a second version by W.W. Breckner [18] in 1978. De…nition 27 Let s 2 (0; 1]: A function f : [0; 1) ! R is named s-convex in the …rst sense if s f ( x + y) for x; y; ; 2 (0; 1) and s + s f (x) + s f (y) = 1: De…nition 28 ([18]) Let s 2 (0; 1]. A function f : [0; 1) ! [0; 1) is named s-convex in the second sense if f ( x + (1 for each x; y 2 (0; 1) and s )y) f (x) + (1 )s f (y) 2 [0; 1]: It can be easily seen that for s = 1, s convexity reduces to ordinary convexity of functions de…ned on [0; 1): Some observations about this kind of functions: a) f is s convex if and only if f ( x + (1 )y) for each x; y 2 (0; 1) and s f (x) + (1 )s f (y) 2 [0; 1]: b) f is convex if and only if f is s convex in any sense. c) if f is s convex in the second sense and f (0) = 0 then f is s convex in the …rst sense. 3.4 m Convex Functions In 1984, G. Toader de…nes the class of m convex functions as follows: De…nition 29 (See [114]) Let f be a real valued function on [0; b]. We will say that it is m convex , where m 2 [0; 1], if we have f (tx + m(1 t)y) tf (x) + m(1 t)f (y) for any x; y 2 [a; b] and t 2 [0; 1]. Also, f is m concave if f is m convex. With Km (b) will denote the class of all m convex functions over [0; b] which f (0) 0. 19 3.5. H CONVEX FUNCTIONS Remark 30 Clearly, 1 convex functions are the classical convex functions, and 0 convex functions are the "starshaped" functions, that is, those functions f that satis…es the inequality f (tx) tf (x), with t 2 [0; 1]. Geometrically a function f : [0; b] ! R is m convex if for any x; y 2 [0; b], say x y , the segment between the points (x; f (x)) and (my; mf (y)) is above the graph of f in [x; my]: Example 31 (See [98]) This is a m convex function for every m 2 (0; 1=2] and discontinuous in x = 1 f (x) = 3.5 1 x 2 3 x 2 1 2 if 0 if 1 x<1 x 2 h Convex Functions A natural way to generalize the concept of Breckner-convexity was proposed by the Croatian mathematician Sanja Varošanec in 2005. De…nition 32 ([115]) Let h : J ! R be a non negative function and h 6 0, de…ned on an interval J R; with (0; 1) J. We shall say that a function f : I ! R , de…ned on an interval I R, is h-convex if f is non negative and the following inequality holds f (tx + (1 t) y) h (t) f (x) + h (1 t) f (y) for any x; y 2 I and for all t 2 (0; 1) : For some results concerning this class of functions see [16, 71, 104]. We can see, from this de…nition, that this class of functions contains the class of Godunova-Levin functions. It also contains the class of 1. If h(t) = 1 then an h convex function f is a P function. 2. If h(t) = ts ; s 2 (0; 1] then an h convex function f is an s function. 3. If h(t) = ts , with s = Levin function. 1 then an h convex function f is a Godunova- Dragomir in [113] introduced an even more general de…nition of operator h-convex functions. 20 CHAPTER 3. GENERALIZED CONVEXITY De…nition 33 Let J be an interval include in R with (0; 1) J. Let h : J ! R be a non negative and identically nonzero function. We shall say that a continuous function f : I ! R , de…ned on an interval I R, is an operator h convex for operators in K if f (tA + (1 t) B) h (t) f (A) + h (1 t) f (B) for all t 2 (0; 1) and A; B 2 K B(H)+ such that Sp(A) I and Sp(B) I; where B(H)+ denote the C algebra commutative of all bounded operators over a Hilbert space H with inner product h; i Sp( ) = ( ) the spectrum of the operator. (See Appendix A) 3.6 Convex Functions As a generalization of the de…nition of convexity, introduced by Gordji M.E. et. al. in his preprint [47], we have the following. De…nition 34 The function f : [a; b] ! R is said to be (or convex respect to ) on [a; b] if the inequality f (tx + (1 t)y) convex function f (y) + t (f (x); f (y)) holds for any x; y 2 I and t 2 [0; 1], and f ([a; b]) ! R: is de…ned by : f ([a; b]) Note that when we choose (x; y) = x y then we are dealing with the classic convex functions. There are some important properties and results about convexity in [47] and [48]. Also in [47], the authors proved some important results but here we give only one of them in the following theorem based on the above de…nition, which is also known as convex version of Hermite-Hadamard inequality. 3.7 GA Convex Functions The following de…nitions are well know in the literature. C. Nicolescu in [86] wrote about the geometric-arithmetic convexity. 21 3.8. (S; M ) CONVEX FUNCTIONS De…nition 35 A function f : I convex if f (xt y (1 t) ) R+ = (0; +1) ! R is said to be GAtf (x) + (1 (3.1) t)f (y) holds for all x; y 2 I and t 2 [0; 1]. Other types of generalized convexity arising from combinations of the above mentioned 3.8 (s; m) Convex Functions In the year 1993 V. Mihesan (see [77]), introduced the class of (s; m) functions as the following: convex De…nition 36 ([77]) The function f : [0; b] ! R is said to be (s; m) convex, where s; m 2 (0; 1], if for every x; y 2 [a; b] and t 2 [0; 1] we have f (tx + m(1 t)y) ts f (x) + m(1 ts )f (y): Note that if s = 1 and m = 1 then (s; m) convexity make match with the classical convexity. Example 37 The function g : [0; 1) ! R de…ned by g(x) = a if x = 0 s bx + d if x > 0 is a (s; 1) convex function for s 2 (0; 1) and a; b; d 2 R with b c a: Example 38 The function g : [0; 1) ! R de…ned by g(x) = is a (1; 16=17) convex. 3.9 1 12 (x4 0 and 5x3 + 9x2 (m; h1; h2) Convex function In [107], Shi D-P, Xi B-Y and Qi F., introduced the following de…nition. De…nition 39 Let h1 ; h2 : [0; 1] ! R+ and m 2 [0; 1]. A function f : [0; 1) ! R is said to be (m; h1 ; h2 ) convex function if f (tx + (1 t)y) h1 (t)f (x) + mh2 (t)f (y) holds for all x; y 2 I and t 2 [0; 1]. If the inequality is reversed is said to be (m; h1 ; h2 ) convave function. 5x) 22 3.10 CHAPTER 3. GENERALIZED CONVEXITY (m; h1; h2) GA Convex function In the year 2016, Bo-Yaw Xi and Fend Qi in [131], introduced the following de…nition: De…nition 40 Sea hi : [0; 1] ! R0 ; m : [0; 1] ! (0; 1] such that hi 6 i=1,2, and f : (0; b] ! R0 . If f (xt y (1 t)m(t) ) h1 (t)f (x) + m(t)h2 (1 t)f (y) 0 for (3.2) for x; y 2 [0; b) and t 2 [0; 1], then f is said to be an (h1 ; h2 ; m) geometricarithmetically convex function or, simply speaking an (h1 ; h2 ; m) GA-convex function. Example 41 Let f (x) = jLn(x)j for x 2 (0; 1], m(t) = c(1 t)l0 for t 2 (0; 1) and 0 < c 1, and some l0 2 R. Let h1 (t) = tl1 and h2 (t) = tl2 for t 2 (0; 1) and l1 ; l2 2 R if l1 ; l2 1, then f is an decreasing and (h1 ; h2 ; m)-GA-convex function on (0; 1]. And f is not an (h; m)-convex function on (0; 1]. 3.11 Relative m-logarithmically- semi-convex function In [89], Noor introduced a new class of convex set and convex function with respect to an arbitrary function; which are called relative convex set and relative convex function respectively, and in [90] established some Hadamard’s type inequality for relative convex functions. Also, the same author have written about this topic in [92]. Let K be a nonempty closed set in a real Hilbert spaces H. De…nition 42 (See [89]) Let Kg be any set in H. The set Kg is said to be relative convex (g-convex) with respect to an arbitrary function g : H ! H such that (1 t)u + tg(v) 2 Kg , 8u; v 2 H : u; g(v) 2 Kg , t 2 [0; 1]. If M is a relative convex set, then it may not be a classical convex set. For example, for M = [ 1; 12] [ [0; 1]and g(x) = x2 ; 8x 2 R. Clearly, this is a relative convex set but not a classical convex set. 3.11. RELATIVE M -LOGARITHMICALLY- SEMI-CONVEX FUNCTION23 De…nition 43 (See [89]) A function f : Kg ! H is said to be relative convex, if there exists an arbitrary function g : H ! H such that f ((1 t)u + tg(v)) (1 t)f (u) + tf (g(v)); 8u; v 2 H : u; g(v) 2 Kg , t 2 [0; 1]: De…nition 44 A function f : Kg ! [0; +1) is a semi logarithmically to be relative convex function with respect to g : H ! H such that Kg is a relative convex set, if f ((1 t)u + tg(v)) (f (u))t (f (v))1 t ; 8u; v 2 H : u; g(v) 2 Kg , t 2 (0; 1). Now, we combine de…nitions of Noor-convexity (relative convexity), mconvexity and semi m-logarithmically-convex for we obtain the class of mlogarithmically- semi-convex functions, as the following. De…nition 45 A function f : Kg ! (0; +1) is: Relative m-logarithmicallysemi-convex function with respect to g : H ! H such that Kg is a relative convex set, if f (m(1 t)u + tg(v)) (f (u))m(1 t) (f (v))t ; (3.3) 8u; v 2 H : u; g(v) 2 Kg , t; m 2 (0; 1): If this inequality reverses, then we call f m-logarithmically semi-concave function. Remark 46 1. If we take m = 1 in (3.3), then we have the de…nition of semilogarithmically-convex function . 2. If we take g(x) = x, in (3.3), then we have the de…nition of (s,m)logarithmically convex function . 24 3.12 CHAPTER 3. GENERALIZED CONVEXITY Other de…nitions regarding generalized convexity In this section we show other concepts about generalized convexity, no less important and useful, that will not be treated in depth in this work and we refer the reader to some articles published about them. 3.12.1 Quasi convex Functions. This section is devoted to the study of some properties of quasi-convex functions, which are used in problems of operations research, quasi-convex programming, game theory, industrial organization, general theory of balance, models of decision making, etc. In [19] and [28] an extensive study of this subject is made. De…nition 47 A function f : D f (tx + (1 t)y) X ! R is said to be quasi convex if max ff (x); f (y)g , t 2 (0; 1) (3.4) and is said to be quasi concave if the inequality (3.4) is reversed. Example 48 Let f : [ 2; 2] ! R de…ned by f (x) = p jxj if 2 x<0 x if 0 x 2 2 is quasi convex but not convex and monotone function. 3.12.2 Wright convex functions E. W. Wright in 1954 (See ) introduced the following de…nition. De…nition 49 A function f : D f (tx + (1 t)y) + f (ty + (1 Rn ! R is Wright convex if t)x) f (x) + f (y); x; y 2 D; t 2 (0; 1) : 3.12. OTHER DEFINITIONS REGARDING GENERALIZED CONVEXITY25 3.12.3 Strongly convex functions In 1966 the Russian mathematician B. T. Polyak in [210] studied the concept of strongly convex function with module c> 0, as described in the next lines. De…nition 50 Let c > 0 and (X; k k) be a normed linear space. A function f : D X ! R is said to be strongly convex with c if f (tx + (1 t)y) tf (x) + (1 t)f (y) + ct(1 t) kx yk2 for all x; y 2 D and t 2 (0; 1) : 3.12.4 Strongly (s; m) convex functions This de…nition appears in the work of Vivas,M. et.al. in [118]. De…nition 51 A function f : [0; 1) ! R is said to be strongly (s; m) convex functions with modulus c in second sense, where (s; m) 2 (0; 1]2 , if f (tx + m(1 t)y) ts f (x) + m(1 t)s f (y) ct(1 t) jx yj2 holds for all x; y 2 [0; 1) and t 2 [0; 1]. Example 52 The function f (x) = g(x) + cx2 is strongly (s; 1) convex , where g is the function de…ned in Example 37 . Example 53 The function f (x) = g(x)+(9=12 + c) x2 is strongly (1; 16=17) convex , where g is the function de…ned in Example 38 . 3.12.5 Relative Strongly h Convex Functions Noor in [91] introduced the class of relative h-convex functions and also discussed some special cases, in addition established some Hermite-Hadamard type inequalities related to relative h-convex functions. De…nition 54 ([91]) A function f : Kg ! H is said to be relative hconvex function with respect to two functions h : [0; 1] ! (0; +1) and g : H ! H such that Kg is a relative convex set, if f ((1 t)u + tg(v)) h(1 8u; v 2 H : u; g(v) 2 Kg, t 2 (0; 1): t)f (u) + h(t)f (g(v)) 26 CHAPTER 3. GENERALIZED CONVEXITY De…nition 55 Let (X; k k) be a real normed space, D stands for a convex subset of X, h : (0; 1) ! (0; 1) is a given function and c is a positive constant. We say that a function f : D ! R is strongly h-convex with module c if f (tx + (1 t)y) h(t)f (x) + h(1 t)f (y) ct(1 t)(x y)2 (3.5) for all x; y 2 D and t 2 (0; 1). Vivas in [129] introduced the following de…nition. De…nition 56 A function f : Kg ! H is said to be relative strongly h-convex function with module c > 0 with respect to two functions h : [0; 1] ! (0; +1) and g : H ! H such that Kg is a relative convex set, if f ((1 t)u + tg(v)) h(1 t)f (u) + h(t)f (g(v)) ct(1 t)ku g(v)k2 (3.6) 8u; v 2 H : u; g(v) 2 Kg, t 2 (0; 1): Remark 57 1. If we take h(t) = t in (3.6), then we have the de…nition of relative strongly convex function with module c. 2. If we take h(t) = ts in (3.6), then the de…nition of relative strongly h-convex function with module c reduces to the de…nition of relative strongly s-convex function with module c. 3. If we take h(t) = 1 in (3.6), then we have the de…nition of relative strongly P -convex function with module c. 4. If we take g(x) = x in (3.6), then we have the de…nition of strongly h-convex function. 5. If we take g(x) = x, h(t) = t in (3.6), then we have the de…nition of strongly convex function with module c. 3.12.6 Invex Convex Functions It is known that if a function f is di¤erentiable in a convex open set D then f is convex it if and only if f (y) f (x) rf (x)(x Rn y) for all y 2 D: The name invex function was given by B. D. Craven in 1981 (See [24]). 3.12. OTHER DEFINITIONS REGARDING GENERALIZED CONVEXITY27 De…nition 58 Let f : D Rn ! R be a di¤erentiable function. We shall say that f is invex respect a function : Rn Rn ! R if f (y) f (x) rf (x) (x; y) for all x; y 2 D: All convex di¤erentiable functions f : D respect to (x; y) = x y: 3.12.7 Rn ! R are invex functions convex function In [116], Veselý and Zajišec have a very extensive treatment of this kind of generalized convexity. De…nition 59 Let X; Y be normed linear spaces, A X an open convex set and F : A ! Y be a mapping. We shall say that F is a convex mapping (or delta-convex mapping) on A if there exist a continuous convex function f on A such that (f + y F ) is a continuous convex function on A for any functional y 2 Y with ky k = 1: We shall say that F is controlled by f or F is a delta-convex mapping with a control function f: The above mentioned authors wrote, about this kind on convexity, the following: "We show that delta-convex mappings have many good di¤erentiability properties of convex functions and the class of them are very stable...some operators which occur naturally in the theory of integral and di¤erential equations are shown to be deltaconvex" 28 CHAPTER 3. GENERALIZED CONVEXITY Chapter 4 Inequalities and Generalized Convexity 4.1 Introduction Many researchers have provided new results regarding the generalized convexity applied to inequalities. In general, the h-convexity, m-convexity, sconvexity, GA-convexity, and combinations of these have been used to treat inequalities like Hermite-Hadamard, Ostrowski, Chebychev, and others, in environments like Numerical Analysis Processes Stochastics and self-adjoint operators in Hilbert spaces. Moreover, in Vector and Normed Spaces , as well as in Banach Spaces have found a relevant position (See [34]) In this chapter we will expose some new results about this topic. 4.2 For P Convex Functions Next we expose the Hermite-Hadamard inequality for P convex function. (See [31]). Theorem 60 Let f 2 P (I),a; b 2 I with a < b and f 2 L ([a; b]) : Then f a+b 2 2 b a Z b f (x)dx 2 (f (a) + f (b)) : a B-Y, Xi et.al. in [133] state the following result in 2012. 29 30 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY Theorem 61 Let f : I R ! R be di¤erentiable on int(I), a; b 2 int(I) 000 with a < b and f 2 L ([a; b]) : If jf 000 jq is P convex on [a; b] for q 1; then 1 b a Z b f (x)dx a (b f a+b 2 (b a)2 00 f 24 a)3 q q jf 000 (a)j + jf 000 (b)j 192 1=q a+b 2 : Some other estimates for the Hermite-Hadamard inequality using P convex functions can be found in the aforementioned cite. 4.3 For s Convex Functions In [35], Dragomir and Fitzpatrick proved a variant of Hermite–Hadamard inequality which holds for the s-convex functions in the second sense. Theorem 62 Suppose that f : [0; 1) ! [0; 1) is a s convex function in second sense, where s 2 (0; 1] ; and let a; b 2 [0; 1) with a < b: If f 2 L1 ([a; b]) then the following inequality holds Z b 1 f (a) + f (b) a+b s 1 f (x)dx : 2 f 2 b a a s+1 The following is known as the Simpson’s inequality for s convex functions in second sense. (Sarikaya, See [105]). Theorem 63 Let f : I [0; 1) ! R be a di¤erentiable function on int(I) and f 0 2 L1 ([a; b]) where a; b 2 int(I) with a < b. If jf 0 j is a s convex function on [a; b] ; for some …xed s 2 (0; 1] ;then the following inequality holds Z b 1 a+b 1 f (x) dx f (a) + f (b) + 4f 6 2 b a a (b a) (s 4) 6s+1 + 2 5s+2 2 3s+2 + 2 (jf 0 (a)j + jf 0 (b)j) 6s+2 (s + 1) (s + 2) In [1], Alomari et al. proved the following inequality of Ostrowski type for functions whose derivative in absolute value are s convex in the second sense. 31 4.4. FOR H CONVEX FUNCTIONS Theorem 64 Let f : I [0; 1) ! R be a di¤erentiable function on int(I) such that f 0 2 L[a; b], where a; b 2 I with a < b. If jf 0 jq is s convex in second sense on [a; b] for some …xed s 2 (0; 1] ; q 1, and jf 0 (x)j M; x 2 [a; b], then we have the following inequality # " Z b 1=q 1 2 (x a)2 + (b x)2 f (x) M f (t)dt b a a s+1 2 (b a) for each x 2 [a; b] : 4.4 4.4.1 For h Convex Functions In the Self adjoint operators in Hilbert Spaces In Appendix A we …nd some basic knowledge about the self adjoint operators in Hilbert Spaces. In the self-adjoint operators in Hilbert space environment, we have the following results, all compiled from the work of Vivas, M and Hernández, J. (See [124],[126]). Theorem 65 (See [124])Let J be an interval include in R with (0; 1) J. Let h : J ! R be a non negative and identically nonzero and supermultiplicative function. Let t1 ; :::; tn be positive real numbers and f : I ! R be an operator h convex function de…ned over an interval I [0; 1) for operators in ,K B(H)+ , and A1 ; :::; An 2 K with (Ai ) I,(i = 1; :::; n) then f where Tn = Pn n 1 X ti Ai Tn i=1 ! n X i=1 h ti Tn f (Ai ) (4.1) i=1 ti Proof. We prove this result by mathematical induction over n 2. If n = 2, the desired inequality is obtained from the De…nition 33 of operator h convex function with t = Tt12 , (1 t) = Tt22 , x = A1 y y = A2 . Assume that for n 1, where n is any positive integer, the inequality (4.1) is also true. 32 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY Then, we see that n 1 X ti Ai Tn i=1 f ! n 1 1 X tn ti Ai An + Tn Tn i=1 = f ! n 1 Tn 1 X ti tn An + Ai Tn Tn i=1 Tn 1 = f ! Again, using the De…nition 33 in the right side of the previous inequality, we have ! ! n 1 n X Tn 1 ti tn 1 X f (An ) + h f ti Ai h Ai f Tn i=1 Tn Tn T n 1 i=1 Now, as we have assumed that (4.1) holds for n 1 we obtain ! n 1 n Tn 1 X ti tn 1 X f (An ) + h h ti Ai h f Tn i=1 Tn Tn Tn i=1 = h tn Tn f (An ) + n 1 X h i=1 Tn 1 Tn h f (Ai ) 1 ti Tn f (Ai ) 1 Further, since h is a supermultiplicative function, we see h Tn 1 Tn h ti Tn using this fact we obtain ! n 1 X ti Ai f Tn i=1 h 1 h = n X i=1 And the proof is complete. tn Tn h Tn 1 ti Tn Tn f (An ) + =h 1 n 1 X i=1 ti Tn Theorem 66 (See [124]) If f is an h and x1 ; :::; xn lie in its domain then " n !# n X X f (xi ) f (xi ) f n n=1 n=1 h ti Tn ti Tn f (Ai ) f (Ai ) : convex function with h(1=2) 6= 0 33 4.4. FOR H CONVEX FUNCTIONS j1 h(1=n)j 2h(1=2) x1 + x2 2 f xn +f x1 + x2 2 +f x2 + x3 2 + 2h n X xn + x1 2 +f : n X 1 2 xn + x1 2 +f f (xi ) n=1 n n h(1=n) X f (xi ) h(1=n)j n=1 " n X 1 f (xi ) h(1=n)j n=1 " n X 1 f (xi ) h(1=n)j n=1 1 j1 j1 j1 therefore " n X 2h 12 f (xi ) j1 h(1=n)j n=1 + xn 2 1 h(1=n)j X f (xi ) h(1=n)j n=1 j1 f (xi ) = j1 n=1 = xn +f but also f n X f (xi ) n n=1 x1 + x2 x2 + x3 +f 2 2 The proof is completed. f + xn 2 1 Note that applying the h convexity property of f we have Proof. f + + h f 1 n n X f (xi ) n=1 n X f (xi ) n=1 n !# # !# +f xn + xn 2 1 +f xn + x1 2 : Corollary 67 Let f : I ! R be an operator h convex function on the interval I [0; 1) for operators in K B(H)+ : Then for all operators Ai 2 K ,(i = 1; ::; n), with spectra in I we have the inequality " n !# n X X 2h 12 f (Ai ) f (Ai ) f j1 h(1=n)j n=1 n n=1 34 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY f A 1 + A2 2 An + :: + f + An 2 1 +f A n + A1 2 Theorem 68 (See [124]) If Let J be an interval include in R with (0; 1) J. Let h : J ! R be a non negative and identically nonzero, supermultiplicative function such that h(t) 1, t 2 (0; 1). If f is an h function and a1 ; :::; an lie in its domain then " n !# n X X 1 f (ai ) f (b1 ) + :: + f (bn ) f (ai ) f j1 h(1=n)j n=1 n n=1 where a= n X ai i=1 Proof. Putting a= n n X ai i=1 we see that n bi = and bi = na n ai ; 1 and bi = na n ai ; (i = 1; ::; n) 1 n X 1 aj j=1;j6=i n ; (i = 1; ::; n) : Since h is supermultiplicative and h(t) h convexity property of f , we have f (b1 ) + :: + f (bn ) h n 1 1 = nh n h(n)h h n X j=1 n X 1 1 n j=1 1 1 n X f (aj ) f (aj ) + :: + h f (aj ) n n 1; t 2 (0; 1) and applying the j=1;j6=1 n X 1 j=1 (i = 1; ::; n) : n X f (aj ) j=1 f (aj ) 1 n 1 n X j=1;j6=n f (aj ) 35 4.4. FOR H CONVEX FUNCTIONS On the other hand we have n X j1 f (aj ) = j1 j=1 1 j1 = j1 j1 n h(1=n)j X f (ai ) h(1=n)j n=1 n h(1=n) X f (ai ) h(1=n)j n=1 " n X 1 f (ai ) h(1=n)j n=1 " n X 1 f (ai ) h(1=n)j n=1 h f n X 1 n f (ai ) n=1 n X f (ai ) n=1 n !# # : So f (b1 ) + :: + f (bn ) j1 " n X 1 f (ai ) h(1=n)j n=1 n X f (ai ) f n=1 n !# : This complete the proof. Corollary 69 Let J be an interval include in R with (0; 1) J. Let h : J ! R be a non negative and identically nonzero and integrable function. Let f : I ! R be an operator h convex function on the interval I [0; 1) + for operators in K B(H) : Then for all operators Ai 2 K ,(i = 1; ::; n), with spectra in I we have the inequality f (B1 ) + :: + f (Bn ) where A= n X Ai i=1 n j1 " n X 1 f (Ai ) h(1=n)j n=1 and Bi = nA n Ai ; 1 f n X f (Ai ) n=1 n !# (i = 1; ::; n) For the next Theorem we use a result proved by Taghavi in [113] which establish the Hermite-Hadamard inequality for self-adjoint operators in Hilbert spaces. 36 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY Let f be an operator h convex function. Then Z 1 1 A+B f (tB + (1 t)A)dt f 2h(1=2) 2 0 Z 1 h(t)dt (f (A) + f (B)) Theorem 70 (4.2) 0 Theorem 71 (See [126]) Let J be an interval include in R with (0; 1) J. Let h : J ! R be a non negative and identically nonzero and integrable function, with h(1=2) 6= 0. Let f : I ! R be an operator h convex function on the interval I [0; 1) for operators in K B(H)+ : Then for all operators A; B 2 K with spectra in I we have the inequality 1 2h(1=2) Z (1 )f (1 )A + (1 + ) B 2 + f (2 )A + B 2 1 f (1 t) A + tB)dt 0 (f ((1 ) A + B) + f (A) + (1 [(h (1 ) + ) f (A) + (h( ) + 1 ) f (B)) ) f (B)] Z Z 1 h(t)dt 0 1 h(t)dt 0 Proof. Since f is an h convex function on the interval I operators in K B(H)+ and by Theorem 70 we have 1 f 2h(1=2) (1 ) A + (1 + ) B 2 Z 1 f (1 t) ((1 ) A + B) + tB)dt 0 (f ((1 ) A + B) + f (B)) Z 0 and 1 f 2h(1=2) (2 )A + B 2 1 h(t)dt [0; 1) for 37 4.4. FOR H CONVEX FUNCTIONS Z 1 f (1 t) A + t ((1 ) A + B))dt 0 (f (A) + f ((1 ) A + B)) Z 1 h(t)dt: 0 Multiplying the …rst of these by (1 adding the inequalities, we obtain 1 f 2h(1=2) (1 ) > 0 and the second by ) A + (1 + ) B + f 2 2h(1=2) (1 ) Z (2 > 0, and )A + B 2 1 f (1 t) ((1 ) A + B) + tB)dt 0 + Z 1 f (1 t) A + t ((1 ) A + B))dt 0 (1 ) (f ((1 ) A + B) + f (B)) Z 1 h(t)dt 0 + (f (A) + f ((1 ) A + B)) Z 1 h(t)dt; 0 and so applying the h convexity property of f , we can reach the desired inequality showed in Theorem. This complete the proof. For Hermite-Hadamard-Fejer type inequality we have. Theorem 72 (See [126]) Let J be an interval include in R with (0; 1) J. Let h : J ! R be a non negative and identically non-zero and integrable function. Let f : [a; b] ! R be an operator h convex function on the interval I [0; 1) for operators in K B(H)+ and g : [a; b] ! R be a non-negative and symmetric function respect to (a + b) =2: Then Z 1 f (tA+(1 t)B)g(tA+(1 t)B)dt (f (A)+f (B)) 0 for all operators A; B 2 K with spectra in [a; b] : Z 0 1 h(t)g(tA+(1 t)B)dt: 38 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY For any A; B 2 K let consider Proof. [A; B] = fZ 2 X : Z = tA + (1 t)B; t 2 [0; 1]g : Let t 2 [0; 1]. We can see that f (tA+(1 t)B)g(tA+(1 t)B) (h(t)f (A) + h(1 f ((1 t)A+tB)g(tA+(1 t)B) (h(1 t)f (B)) g(tA+(1 t)B); t)f (A) + h(t)f (B)) g((1 t)A+tB): After adding and integrate both inequalities we get Z 1 Z 1 f ((1 t)A+tB)g((1 t)A+tB)dt f (tA+(1 t)B)g(tA+(1 t)B)dt+ 0 0 Z 1 (h(t)f (A)g(tA + (1 t)B) 0 + h(1 t)f (B)g(tA + (1 + h(1 t)B) t)f (A)g((1 t)A + tB) + h(t)f (B)g((1 = Z t)A + tB)) dt 1 (f (A) [h(t)g(tA + (1 t)B) + h(1 t)g((1 t)A + tB)] 0 + f (B) [h(1 t)g(tA + (1 t)B) + h(t)g((1 t)A + tB)]) dt since g is symmetric respect (a + b) =2 we have Z Z 1 h(1 t)g((1 t)A + tB)dt = 1 h(t)g(tA + (1 t)B)dt 0 0 and therefore Z Z 1 f (tA+(1 t)B)g(tA+(1 t)B)dt+ 1 f ((1 t)A+tB)g((1 t)A+tB)dt 0 0 2f (A) Z 1 h(t)g(tA + (1 t)B)dt 0 + 2f (B) Z 0 1 h(t)g(tA + (1 t)B)dt 39 4.4. FOR H CONVEX FUNCTIONS = 2(f (A) + f (B)) Z 1 h(t)g(tA + (1 t)B)dt 0 and with an appropriate substitution in the left hand term Z 1 f (tA + (1 t)B)g(tA + (1 t)B)dt 0 Z 1 h(t)g(tA + (1 t)B)dt: (f (A) + f (B)) 0 Theorem 73 (See [126]) Let h : [0; maxf1; b ag] ! R be a non negative and identically non-zero and integrable function. Let f : [a; b] ! R be an operator h convex function on the interval I [0; 1) for operators in K B(H)+ and g : [a; b] ! R be a non-negative and symmetric operator function respect to (a + b) =2: Then f A+B 2 R1 0 2h(1=2) g (tA + (1 1 [f (tA + (1 t)B) g (tA + (1 t)B)] dt: 0 Using the h convexity of f; we have Proof. f t)B) dt Z t)A + tB + (1 t)B 2 h(1=2) [f (tA + (1 t)B) + f (tB + (1 A+B 2 = f tA + (1 t)A)] : Since g is positive and symmetric respect (A + B) =2 g (tA + (1 t)B) f A+B 2 h(1=2) [f (tA + (1 t)B) g (tA + (1 +f (tB + (1 t)B) t)A) g (tB + (1 t)A)] and integrating f A+B 2 R1 0 2h(1=2) g (tA + (1 t)B) dt Z 0 1 [f (tA + (1 t)B) g (tA + (1 t)B)] dt: 40 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY Theorem 74 (See [126]) Let J be an interval include in R with (0; 1) J. Let h1 ; h2 : J ! R be two non negative, identically non-zero, (h1 ; h2 ) 2 L1 (J): Let f : I ! R be an operator h1 convex and g : I ! R be an operator h2 convex functions for operators in K B(H)+ with spectra in I: Then Z 1 h(f (tB + (1 t)A)) x; xi h(g(tB + (1 t)A)) x; xi dt 0 M (A; B) Z 1 h1 (t)h2 (t)dt + N (A; B) 1 h1 (t)h2 (1 t)dt 0 0 where Z M (A; B) = hf (A)x; xi hg(A)x; xi + hf (B)x; xi hg(B)x; xi and N (A; B) = hf (A)x; xi hg(B)x; xi + hf (B)x; xi hg(A)x; xi : Proof. For x 2 H with kxk = 1 and t 2 [0; 1] we have h(tA + (1 t) B) x; xi = t hAx; xi + (1 t) hBx; xi 2 I Since hAx; xi 2 Sp (A) I and hBx; xi 2 Sp (B) I . Continuity of f; g and the previous equality imply that the following operator valued integrals exists Z 1 f (tB + (1 t)A)dt; Z 0 1 g(tB + (1 t)A)dt 0 and Z 1 f (tB + (1 t)A)g(tB + (1 t)A)dt: 0 For t 2 [0; 1] ; by the h convexity property of each one, we have thus h(f (tB + (1 t)A)) x; xi h1 (t) hf (A)x; xi + h1 (1 t) hf (B)x; xi ; h(g(tB + (1 t)A)) x; xi h2 (t) hg(A)x; xi + h2 (1 t) hg(B)x; xi h(f (tB + (1 t)A)) x; xi h(g(tB + (1 t)A)) x; xi 41 4.4. FOR H CONVEX FUNCTIONS h1 (t)h2 (t) hf (A)x; xi hg(A)x; xi + h1 (1 t)h2 (1 t) hf (B)x; xi hg(B)x; xi + h1 (t)h2 (1 t) hf (A)x; xi hg(B)x; xi + h1 (1 t)h2 (t) hf (B)x; xi hg(A)x; xi integrating both sides of the last inequality Z 1 h(f (tB + (1 t)A)) x; xi h(g(tB + (1 t)A)) x; xi dt 0 hf (A)x; xi hg(A)x; xi Z 1 h1 (t)h2 (t)dt 0 + hf (B)x; xi hg(B)x; xi + hf (A)x; xi hg(B)x; xi + hf (B)x; xi hg(A)x; xi but Z Z Z 1 h1 (1 t)h2 (1 h1 (t)h2 (1 Z Z 0 thus we obtain Z 1 h(f (tB + (1 0 t)dt 1 h1 (1 t)h2 (t)dt 0 t)dt = Z 1 h1 (s)h2 (s)ds 0 0 and t)dt 0 1 h1 (1 t)h2 (1 0 1 1 h1 (1 t)h2 (t)dt = Z t)A)) x; xi h(g(tB + (1 1 h1 (s)h2 (1 s)dt 0 t)A)) x; xi dt (hf (A)x; xi hg(A)x; xi + hf (B)x; xi hg(B)x; xi) Z 1 h1 (t)h2 (t)dt 0 +(hf (A)x; xi hg(B)x; xi + hf (B)x; xi hg(A)x; xi) which can be written like Z 1 h(f (tB + (1 t)A)) x; xi h(g(tB + (1 0 t)A)) x; xi dt Z 0 1 h1 (t)h2 (1 t)dt 42 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY M (A; B) Z 1 h1 (t)h2 (t)dt + N (A; B) 1 h1 (t)h2 (1 t)dt 0 0 where Z M (A; B) = hf (A)x; xi hg(A)x; xi + hf (B)x; xi hg(B)x; xi and N (A; B) = hf (A)x; xi hg(B)x; xi + hf (B)x; xi hg(A)x; xi : Theorem 75 (See [126] ) Let J be an interval include in R with (0; 1) J. Let h1 ; h2 : J ! R be two non negative, identically non-zero, (h1 ; h2 ) 2 L1 (J). Let f : I ! R be an operator h1 convex and g : I ! R be an operator h2 convex functions for operators in K B(H)+ with spectra in I: Then for all operators with spectra in I 1 2h1 1 2 h2 Z 1 2 f( A+B )x; x 2 g A+B 2 x; x 1 0 hf (tA + (1 t)B) x; xi hg (tA + (1 +2 M (a; b) Z t)B) x; xi 1 h1 (t)h2 (1 t)dt + N (a; b) 1 h1 (t)h2 (t)dt 0 0 where Z M (a; b) = hf (A)x; xi hg(A)x; xi + hf (B)x; xi hg(B)x; xi and N (a; b) = hf (A)x; xi hg(B)x; xi + hf (B)x; xi hg(A)x; xi for for any x 2 H with kxk = 1: First we note that Proof. f( A+B )x; x 2 = f g( A+B )x; x 2 = g tA + (1 t) A + tB + (1 2 t)B tA + (1 t) A + tB + (1 2 t)B x; x and x; x 43 4.4. FOR H CONVEX FUNCTIONS then we can observe that f( = A+B )x; x 2 f g tA + (1 1 2 h2 x; x t) A + tB + (1 2 g h1 A+B 2 1 2 tA + (1 1 2 h2 (hf (tA + (1 + hf ((1 1 2 x; x t) A + tB + (1 2 t)B) x; xi + hg ((1 f[hf (tA + (1 (h2 (1 t) A + tB) x; xi) t)B) x; xi t) hf (B)x; xi) t) hg(A)x; xi + h2 (t) hg(B)x; xi) t) hf (A)x; xi + h1 (t) hf (B)x; xi) (h2 (t) hg(A)x; xi + h2 (1 1 2 t) A + tB) x; xi) t) A + tB) x; xi] + (h1 (t) hf (A)x; xi + h1 (1 h1 x; x t)B) x; xi hg (tA + (1 t) A + tB) x; xi hg ((1 + (h1 (1 t)B t)B) x; xi + hf ((1 (hg (tA + (1 h1 t)B h2 t) hg(B)x; xi)g 1 2 f[hf (tA + (1 t)B) x; xi hg (tA + (1 + hf ((1 + (h1 (t)h2 (1 t)B) x; xi t) A + tB) x; xi hg ((1 t) + h1 (1 t) A + tB) x; xi] t)h2 (t)) (hf (A)x; xi hg(A)x; xi + hf (B)x; xi hg(B)x; xi) + (h1 (t)h2 (t) + h1 (1 t)h2 (1 t)) (hf (A)x; xi hg(B)x; xi + hf (B)x; xi hg(A)x; xi)g 44 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY Now integrating over [0; 1] we have )x; x g f ( A+B 2 2h1 21 h2 Z A+B 2 1 2 x; x 1 0 hf (tA + (1 + 2M (a; b) t)B) x; xi hg (tA + (1 Z t)B) x; xi dt 1 h1 (t)h2 (1 t)dt + 2N (a; b) 1 h1 (t)h2 (t)dt 0 0 where Z M (a; b) = hf (A)x; xi hg(A)x; xi + hf (B)x; xi hg(B)x; xi and N (a; b) = hf (A)x; xi hg(B)x; xi + hf (B)x; xi hg(A)x; xi This complete the proof. Theorem 76 (See [126] Let J be an interval include in R with (0; 1) J. Let h1 ; h2 : J ! R be two non negative, identically non-zero, (h1 ; h2 ) 2 L1 (J). Let f : I ! R be an operator h1 convex and g : I ! R be an operator h2 convex functions for operators in K B(H)+ with spectra in I: Then for all operators with spectra in I f( A+B )x; x 2 where g A+B 2 2 (M (A; B) + N (A; B)) Z 1 t h2 h1 2 0 x; x t 2 dt + Z 0 1 h1 t 2 h2 1 t 2 dt M (A; B) = hf (A)x; xi hg(A)x; xi + hf (B)x; xi hg(B)x; xi and N (A; B) = hf (A)x; xi hg(B)x; xi + hf (B)x; xi hg(A)x; xi : 45 4.4. FOR H CONVEX FUNCTIONS First we note that applying h1 convexity Proof. f( A+B )x; x 2 tA + (1 = f h1 t 2 f (A) + h1 +h1 = h1 t 2 + h1 1 t) A + tB + (1 2 t t 2 1 t t)B x; x f (A) 2 1 f (B) + h1 t f (B) x; x 2 (hf (A)x; xi + hf (B)x; xi) 2 and using the h2 convexity g( A+B )x; x 2 = tA + (1 g t 2 h2 g(A) + h2 +h2 = h2 t 2 + h2 1 t 2 t) A + tB + (1 2 t 2 1 t t)B x; x g(A) 2 g(B) + h2 1 t 2 g(B) x; x (hg(A)x; xi + hg(B)x; xi) and with these f( = = A+B )x; x 2 h1 h1 t 2 t 2 + h1 g A+B 2 1 t 2 x; x h2 t 2 + h2 1 t 2 (hf (A)x; xi + hf (B)x; xi) (hg(A)x; xi + hg(B)x; xi) + h1 1 t 2 h2 t 2 + h2 1 t 2 (hf (A)x; xi hg(A)x; xi + hf (B)x; xi hg(B)x; xi 46 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY + hf (A)x; xi hg(B)x; xi + hf (B)x; xi hg(A)x; xi) integrating over [0; 1] we have f( A+B )x; x 2 g A+B 2 x; x (M (A; B) + N (A; B)) Z 1 1 t t 1 t t + h1 h2 + h2 h1 2 2 2 2 0 = 2 (M (A; B) + N (A; B)) Z 1 Z 1 t t t 1 t h1 h2 dt + h2 dt h1 2 2 2 2 0 0 where dt M (A; B) = hf (A)x; xi hg(A)x; xi + hf (B)x; xi hg(B)x; xi and N (A; B) = hf (A)x; xi hg(B)x; xi + hf (B)x; xi hg(A)x; xi and this complete the proof. Also we can …nd some results about re…nement of the Hemite-Hadamard Inequality, in the work of Vivas, M. and Hernández, J. (See [126]). Theorem 77 Let f : I ! R be an operator h convex function on some interval I. Then for any self-adjoint operators A and B with spectra in I, we have the inequality f k 1 1X f k i=0 A+B 2 Z (2k 2i 1) A + (2i + 1) B 2k 1 f ((1 t) A + tB) dt 0 "k 1 1 X f k i=0 (k ( i) A + iB k + h(1=2) (f (A) + f (B)) h(1=2) (f (A) + f (B))) where k is the numbers of steps. # (4.3) 47 4.4. FOR H CONVEX FUNCTIONS R1 Proof. The function f is continuous, 0 f ((1 t) A + tB) dt exists for any self-adjoint operators A and B with spectra in I. We can give two proofs of the theorem. The …rst using the de…nition of operator h convex functions and the second using the Hermite-Hadamard inequality for real-valued functions. The …rst proof. From the de…nition of operator h convex functions, we have the inequalities f X +Y 2 h =f 1 2 (f ((1 (1 (1 t)X + tY + 2 t)X + tY ) + f ((1 h 1 2 t)Y + tX 2 (4.4) t)Y + tX)) (f (X) + f (Y )) for anyt 2 [0; 1] and self-adjoint operators X and Y with spectra in I. If we integrate the inequality (4.4) over t and take into account that Z 1 f ((1 t) X + tY ) dt = Z 1 f (tX + (1 t) Y ) dt 0 0 then we conclude the Hermite-Hadamard inequality for operator h convex functions f X +Y 2 Z 1 f ((1 t) X + tY ) dt 0 h 1 2 (f (X) + f (Y )) (4.5) that holds for any self-adjoint operators X and Y with spectra in I. Utilizing 48 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY the change of variable u = kt, we have Z 1=k f ((1 t) A + tB) dt 0 Z 1 = k 1 = k 1 = k 1 f u u A + B du k k u u A + B du k k (k 1)A + B u) A + u k 1 0 Z 1 f A 0 Z 1 f (1 0 and by the change of variable u = kt Z 1, we have 2=k f ((1 1=k 1 = k du Z 1 = k 1 = k Z t) A + tB) dt f 1 0 Z 1 f 0 1 f 1 (1 u) (k 0 A u+1 u+1 A+ B du k k Au A Bu B du + + k k k k 1)A + B (k +u k 2)A + 2B k du We can change the variables until the variable u = kt (k 1) by using the same procedure above. By the change of variable u = kt (k 1), we get Z 1 f ((1 t) A + tB) dt (k 1)=k 1 = k 1 = k 1 = k Z 1 f 1 0 Z 1 f A f (1 0 Z 0 1 u+k 1 u+k 1 A+ B du k k Au A Bu B du A+ + +B k k k k A + (k 1)B u) + uB du : k 49 4.4. FOR H CONVEX FUNCTIONS Using the Hermite-Hadamard inequality in (4.5), we have ! A + (k 1)A+B (2k 1) A + B k f =f 2 2k Z 1 f (1 u) A + u (k 0 h (k 1)A+B k f Z 1 2 + (k 2 2)A+2B k 1 f (1 1 2 f (k 2)A+2B k Z (k f 1 f (1 (k u) 0 h 1 2 (k f (2k =f ; 3) A + B 2k (4.7) (k (k 3)A+3B k + 2 ! du 1) A + B du k 1) A + B (k 2) A + 2B +f k k u) A + u 0 h (k f (A) + f 1) A + B k 1) A + B k (4.6) ! (2k =f ; 5) A + 5B 2k (4.8) 2) A + 2B (k 3) A + 3B du +u k k 2) A + 2B (k 3) A + 3B +f k k : : : By induction we have f A+(k 1)B k 2 +B ! =f A + (2k 1)B 2k (4.9) 50 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY Z 1 f (1 0 1 2 h f A + (k 1)B + uB du k A + (k 1)B + f (B) k u) By summing (4.6), (4.7), (4.8), (4.9) and the other inequalities between (4.8) and (4.9), we have ! ! (k 2)A+2B (k 1)A+B + A + (k 1)A+B k k k f +f 2 2 (k 2)A+2B k +f k Z + 2 (k 3)A+3B k ! + :::: + f A+(k 1)B k 2 +B ! 1 f ((1 t) A + tB) dt 0 h 1 2 + f + f f (A) + f (k 1) A + B k (k 1) A + B k +f (k 1) A + B k +f ::: + f (k 2) A + 2B k (k 2) A + 2B k A + (k 1)B k + f (B) (4.10) When regulating the inequality (4.10), we get the desired inequality in Theorem. It is obvious from the left-hand side of the inequality (4.3) for k = 1,we get f ( A+B ), and it is obvious the right-hand side of the inequality (4.3) is 2 provided for k = 2. The second proof. Let x 2 H, kxk = 1 and let A and B be two self-adjoint operators with spectra in I. De…ne the real-valued function 'x;A;B : [0; 1] ! R by 'x;A;B (t) = hf ((1 t)A + tB)x; xi : Since f is an operator h convex, then for any t1 ; t2 2 [0; 1] y ; 0 con + = 1; we have 'x;A;B ( t1 + t2 ) 51 4.4. FOR H CONVEX FUNCTIONS = hf ((1 ( t1 + t2 ))A + ( t1 + t2 )B)x; xi = hf ( [(1 t1 ) A + t1 B] + [(1 t2 ) A + t2 B])x; xi h( ) hf ( [(1 t1 ) A + t1 B]) x; xi +h( ) hf ( [(1 t2 ) A + t2 B]) x; xi = h( )'x;A;B (t1 ) + h( )'x;A;B (t2 ) showing that 'x;A;B is a h convex function on [0; 1] : Now we can use the Hermite-Hadamard inequality for real-valued functions Z b a+b 1 g g(s)ds 2 b a a 1 (g(a) + g(b)) h 2 to get that 'x;A;B 1 2k k Z 1=k 'x;A;B (t)dt 0 1 2 h 3 2k 'x;A;B k Z 'x;A;B (0) + 'x;A;B 1 k ; 2=k 'x;A;B (t)dt 1=k h 1 2 'x;A;B 1 k + 'x;A;B 1 2k ; . . . 'x;A;B 2k 1 2k k Z 1 'x;A;B (t)dt (k 1)=k h 1 2 'x;A;B k 1 k + 'x;A;B (1) : By summing the inequalities above and multiplying with (1=k), we get 1 ' k x;A;B 1 2k + 'x;A;B 3 2k + :: + 'x;A;B 2k 1 2k 52 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY Z 1 'x;A;B (t)dt 0 1 2 1 h k 'x;A;B (0) + 'x;A;B (1) + 'x;A;B +:: + 'x;A;B k 1 k 1 k Thus, we can write 1 k f 2 k 1 2 A+ B k f k 1 Z 1 2 1 A+ k k 1 1 k hf ((1 B A+ 3 B 2k + :: x; x t) A + tB) x; xi dt f (A) + f (B) + f f 3 2k 1 1 0 1 h k +f k 1 k 1 k 1 A+ B k + ::: 1 B x; x : k k By regulating these inequalities above, we get # + *" k 1 X (2k 2i 1) A + (2i + 1) B 1 x; x f k 2k i=0 Z 0 1 h k 1 2 *" A+ 1 1 hf ((1 f (A) + f (B) + k 1 X i=0 f (k i) A + iB k Finally, since by the continuity of the function f , we have Z 1 hf ((1 t) A + tB) x; xi dt 0 (4.11) t) A + tB) x; xi dt # + x; x : 53 4.4. FOR H CONVEX FUNCTIONS = Z 1 f ((1 t) A + tB) dt x; x 0 for any x 2 H, and any two self-adjoint operators A and B with spectra in I, from (4.11) we get the desired result in (4.3). 4.4.2 In the Real Line in Fractal Sets Environment In Appendix C we …nd some basic knowledge about the real Straight in fractal sets. All the results mentioned here are established in the publication of Vivas, M. and Hernández, J in [123]. Theorem 78 Let t1 ; :::; tn be positive real numbers. If h : J ! R is a nonnegative function, h 6 0, supermultiplicative de…ned over an interval J R and such that (0; 1) J, and let f : I ! R be a function de…ned over an interval I R, h convex, and x1 ; :::; xn 2 I, then ! n n X 1 X ti f f (xi ) (4.12) ti xi h Tn i=1 T n i=1 Pn where Tn = i=1 ti . Proof. The proof is by induction. If n = 2, the desired inequality is obtained from the de…nition of h-convex function (32) with t = Tt12 , (1 t) = Tt22 , x = x1 and y = x2 . Assume that for n 1, where n is any positive integer, the inequality (4.12) is also true. Then, we see that ! ! n n 1 1 X tn 1 X ti xi ti xi xn + = f f Tn i=1 Tn Tn i=1 ! n 1 Tn 1 X ti tn = f xn + xi : Tn Tn i=1 Tn 1 Using the de…nition (32) in the right-hand side of the previous inequality, we have ! ! n 1 n X 1 X ti Tn 1 tn f (xn ) + h f xi : ti xi h f Tn i=1 Tn Tn T i=1 n 1 54 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY Now, as we have assumed that (4.12) holds for n 1 we obtain ! n 1 n 1 X Tn 1 X ti tn f f (xn ) + h h h ti xi Tn i=1 Tn Tn Tn 1 i=1 = h tn Tn n 1 X f (xn ) + h i=1 Tn 1 Tn f (xi ) ti h Tn f (xi ) : 1 Further, since h is a supermultiplicative function, we can see h Tn 1 Tn h using this fact we obtain ! n tn 1 X h ti xi f Tn i=1 Tn ti Tn h 1 f (xn ) + n 1 X Tn 1 ti Tn Tn ti Tn h i=1 ti Tn =h 1 f (xi ) = n X i=1 ; h ti Tn f (xi ) : The above inequality holds by the result for n = 2 and the induction hypothesis. The next result involves an integral inequality of Hermite-Hadamard type. Theorem 79 Let h : J ! R be a non-negative integrable function, h 6 0, de…ned over an interval J R and such that (0; 1) J and f : I ! R be an h convex, non-negative and integrable function, a; b 2 I with a < b. Then a+b 1 f( ) ( 1) ) h(1=2) (1 + ) 2 (1 1 (b a) (f (b) ( ) a Ib f (4.13) ( ) ( 1) f (a)) 0 I1 h: Proof. Note that ta + (1 t) b + (1 t) a + tb = ta + b tb + a ta + tb = a + b for all t 2 [0; 1]. And as f is an h convex function, we have f a+b 2 h (1=2) f (ta + (1 = h (1=2) (f (ta + (1 t) b) + h (1=2) f ((1 t) b) + f ((1 t) a + tb) t) a + tb)) : 55 4.4. FOR H CONVEX FUNCTIONS Thus, integrating both sides, we get Z 1 a+b (dt) f 2 0 Z Z 1 f (ta + (1 t) b) (dt) +h (1=2) h (1=2) Now, we note that Z 1 f (ta + (1 0 Z ( 1) t) b) (dt) = (b a) 1 f ((1 f ((1 t) a + tb) (dt) : 0 0 and 1 t) a + tb) (dt) = 0 and with this we have Z 1 a+b f (dt) 2 0 1 (b h (1=2) (1 (b a) a) Z b f (x) (dx) a Z b f (x) (dx) a ( 1) ) Z b f (x) (dx) a from which it follows that (1 1 f ( 1) ) h (1=2) (1 + ) a+b 2 1 (b a) ( ) a Ib f which corresponds to the left inequality in (4.13). We know that for any x 2 [a; b] there exists t 2 [0; 1] such that x = ta + (1 t) b. With this fact and the h convexity of f , we can write Z b Z 1 f (x) (dx) = (b a) f ((1 t) a + tb) (dt) a 0 Z 1 (b a) (h (1 t) f (a) + h (t) f (b)) (dt) 0 Z 1 Z 1 h (t) (dt) h (1 t) (dt) + f (b) = (b a) f (a) 0 0 Z 1 Z 1 h (t) (dt) h (t) (dt) + f (b) f (a) ( 1) = (b a) 0 0 Z 1 h (t) (dt) = (b a) ( ( 1) f (a) + f (b)) 0 56 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY an so we obtain 1 (b a) ( ) ( ) a Ib f ( ( 1) f (a) + f (b))0 I1 h which corresponds to the right-hand side of (4.13), and we can conclude 1 a+b f( ) ( 1) ) h(1=2) (1 + ) 2 (1 1 (b a) ( ) a Ib f ( ) (f (b) ( 1) f (a))0 I1 h: This complete the proof. 4.5 For m Convex Functions The following results appear in a Vivas, M.J. et.al. publication in the year 2016. (See [128]) Theorem 80 Let f : [0; 1) ! R be a m convex function, with m 2 (0; 1], which is integrable over [a; b] where a; b 2 [0; 1) and let g : [a; b] ! R be a non-negative and integrable function which is symmetric respect to (a + b) =2: Then we have Z b Z f (a) + f (b) b b x f (x)g(x)dx g(x)dx 2 b a a a Z b a b x a m f +f g(x)dx + 2 m m b a a Proof. Since g is non-negative and integrable function which is symmetric respect to (a + b) =2 we have Z b a 1 f (x)g(x)dx = 2 1 = 2 1 = 2 Z a b f a b b x a Z b f (x)g(x)dx + b f (a + b x)g(a + b x)dx a a Z Z b (f (x) + f (a + b x)) g(x)dx a +b x b a a +f a b b x a +b x b a a g(x)dx 57 4.5. FOR M CONVEX FUNCTIONS Hence, the m convexity of f implies Z b f (x)g(x)dx a 1 2 Z a b b b x a f (a) + m x b +m f (a) + f (b) = 2 Z a b b b a a x a x b f a a f a + m m g(x)dx+ f 2 b m b b x a b m f (b)g(x)dx a +f m Z a b x b a a g(x)dx Theorem 81 Let f : [0; 1) ! R be a m convex function, with m 2 (0; 1], which is integrable in [a; b] where 0 a < b < 1 and let g : [a; b] ! R be a non-negative and integrable function which is symmetric respect to (a + b) =2: Then we have Z b Z Z 1 b x m b a+b g(x)dx f (x)g(x)dx + f g(x)dx f 2 2 a 2 a m a Proof. The m convexity of f implies Z b Z b a+b a+b+x x f g(x)dx f g(x)dx = 2 2 a a Z b a+b x m x f g(x)dx + f 2 2 m a Z b Z b a+b x m x f = g(x)dx + g(x)dx: f 2 m a a 2 Now, the hypotheses of g imply that this last expression is equal to Z b Z b m x a+b x g(a + b x)dx + g(x)dx f f 2 m a 2 a Z Z 1 b x m b = f (x) g(x)dx + f g(x)dx 2 a 2 a m wich proves the result. We need the following Lemma which is established in [128]. 58 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY Lemma 82 If f : [0; 1) ! R is an m convex function, with m 2 (0; 1], then, for all x 2 [a; b] [0; 1) there is = x 2 [0; 1] such that f (a + b x) m (1 ) f a +f m b m + (f (a) + f (b)) f (x) Theorem 83 Let f : [0; 1) ! R be a m convex function, with m 2 (0; 1], which is integrable over [a; b] where a; b 2 [0; 1) and let g : [a; b] ! R be a non-negative and integrable function which is symmetric respect to (a + b) =2: Then we have Z b Z b m f (a) + f (b) a b f (x)g(x)dx g(x)dx f +f + 2 m m 2 a a Proof. By the symmetry of g with respect to (a + b)=2 and Lemma 82 we have Z b f (x)g(x)dx a 1 = 2 1 = 2 1 2 Z f (a + b x)g(a + b a Z Z b b f (a + b a 1 x)g(x)dx + 2 b m (1 ) f a m a f +f 2 m 4.6 x)dx + Z Z b f (x)g(x)dx a b f (x)g(x)dx a a b +f + (f (a) + f (b)) m m Z 1 b f (x)g(x)dx + 2 a Z b f (a) + f (b) b + g(x)dx m 2 a f (x) g(x)dx For GA Convex Functions We cite here some results from the work of Latif, M.A. (See [72] ). The following Lemma is necessary. 59 4.6. FOR GA CONVEX FUNCTIONS Lemma 84 Let f : I R+ ! R be a di¤erentiable function on int(I) and a; b 2 I with a < b. If f 0 2 L [a; b], then the following equality holds: Z bf (b) af (a) b f (x)dx a = ln b ln a 2 Z 1 b 1+t 1 t 0 a f b 1+t 2 a 1 t 2 dt + Z 1 b1 t a1+t f 0 b 1 t 2 a 1+t 2 dt 0 0 The hermite Hadamard inequality for GA convex functions is obtained by Lati¤, M.A. and we rewrite his proof. Theorem 85 (See Theorem 2.2 in [72]) Let f : I R+ ! R be a di¤erentiable function on int(I) and a; b 2 I with a < b and f 0 2 L [a; b]. If jf 0 jq is GA convex on [a; b] for q 1, we have the following inequality: bf (b) Z af (a) b f (x)dx a (b a)1 1=q n b (L(a; b) 21=q + 1 + a (b q a) jf 0 (a)j + (2b q 1=q q L(a; b)) jf 0 (a)j + (b 2a q 1=q L(a; b)) jf 0 (b)j a L(a; b)) jf 0 (b)j Proof. From Lemma 84 and Hölder’s inequality , we have bf (b) Z af (a) o : b f (x)dx a ab (ln(b) ln(a)) = 2 + 0 1 b a 0 8 Z ln(a)) < ab (ln(b) 2 Z "Z 1 a b t : dt 1 0 1 1=q t f0 b 1+t 2 a 1 t 2 dt + Z 1 0 t b a !1 1=q dt Z 0 1 a b t f0 b Z 1 0 b a 1 t 2 1+t 2 a a b t t f0 b q f0 b 1+t 2 1=q dt a ) : 1 t 2 1 t 2 a q 1+t 2 # dt !1=q dt (4.14) 60 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY By the GA convexity of jf 0 jq and using integration by parts, we have Z 1 t b a 0 f0 b 1+t 2 q 0 jf (a)j q = jf 0 (a)j and Z 1 a b 0 t f0 b Z q 1 dt t b a 1 t 2 q 0 dt + jf (b)j Z 1 0 t b a 1+t 2 dt L(a; b) a a L(a; b) q 2b + jf 0 (b)j 2a (ln b ln a) 2a (ln b ln a) 1 t 2 jf (a)j = jf 0 (a)j q 1 t 2 0 q 0 a 1+t 2 a Z 0 1 q a b (4.15) dt t 1+t 2 0 q dt + jf (b)j Z 0 1 a b t 1 t 2 dt L(a; b) a 2b a L(a; b) q + jf 0 (b)j 2a (ln b ln a) 2a (ln b ln a) (4.16) Using (4.15) and (4.16) in (4.14), we get the required result. The proof is complete. 4.7 For (s; m) Convex Functions Next result establish the Fejér-Hadamard inequality version for (s; m) convex functions, and appeared in the work of Vivas, M.J. (See [117]). Theorem 86 Let f : I [0; +1) ! [0; +1) be a (s; m) sense function, a; b 2 I with a < b, f 2 L1 ([a; b]) and a+b nonnegative, integrable and symmetric about . Then 2 Z Z b s f (a) + f (b) b b x g(x)dx f (x)g(x)dx 2 b a a a Z b m a b + f +f 2 m m a convex in second g : [a; b] ! R is x b a a s g(x)dx: 61 4.7. FOR (S; M ) CONVEX FUNCTIONS Proof. Since f and g are real nonnegative functions, g is integrable and a+b symmetric about , we will have that 2 Z b f (x)g(x)dx a Z b Z b 1 f (a + b x)g(a + b x)dx f (x)g(x)dx + = 2 a a Z 1 b = [f (x) + f (a + b x)]g(x)dx 2 a Z 1 b x a b x = +b f a 2 a b a b a b x x a +b g(x)dx +f a b a b a Z s s 1 b b x x a b f (a) + m f 2 a b a b a m s s x a b x a +m + f f (b) g(x)dx b a m b a Z s f (a) + f (b) b b x g(x)dx = 2 b a a Z b s m a b x a + f +f g(x)dx: 2 m m b a a The proof is complete. Remark 87 Note that if we do m = 1 in the previous theorem we obtain inequality of the Hermite-Hadamard-Féjer type for s-convex functions, i.e.: Z b f (x)g(x)dx a f (a) + f (b) 2 Z a b b b x a s + x b a a s g(x)dx: Corollary 88 Under the same hypotheses of theorem 86, if g(x) = 1, we have: 1 (b a) Z a b f (x) dx a f (a) + f (b) m f +f + 2 2 m b m 1 2s : 62 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY Proof. If we take g(x) = 1 in theorem 86, we have: Z b f (x)dx a Z s f (a) + f (b) b b x dx 2 b a a m a b + f +f 2 m m Z a Given that the function '(x) = xs is concave if 0 < s inequality: 1 b a Z b f (a) + f (b) 2 f (x)dx a 1 b a Z a b b b b x b a a s dx: 1, then from Jensen’s s x dx a Z b x a b 1 m +f f + 2 m m b a a b a 1 f (a) + f (b) m b + f +f = s 2 2 2 m m 1 b a = s+1 f (a) + f (b) + m f +f 2 m m a dx a s : The proof is complete. The following theorem will serve for a lower bound to the left inequality (2.3), for functions (s; m)-convex in the second sense. Theorem 89 Let f : I R ! [0; +1) be is (s; m) convex in second sense function, a; b 2 I with a < b, f 2 L1 ([a; b]) and g : [a; b] ! R is nonnegative, a+b . Then integrable and symmetric about 2 f a+b 2 Z a b g(x)dx 1 2s Z a b m f (x)g(x)dx + s 2 Z a b f x g(x)dx: m Proof. Since f : I R ! [0; +1) be is (s; m) convex in second sense function and g : [a; b] ! R is nonnegative, integrable and symmetric about 4.8. FOR STRONGLY (S; M ) CONVEX FUNCTIONS a+b , we have 2 Z b Z b a+b f f g(x)dx = 2 a a Z b a 1 2s 1 = s 2 1 = s 2 = a+b x+x g(x)dx 2 1 f (a + b 2s 63 x m f 2s m g(x)dx Z b Z b x m f f (a + b x) g(x)dx + s g(x)dx 2 m a a Z b Z b m x f (a + b x) g(a + b x)dx + s g(x)dx f 2 m a a Z b Z x m b g(x)dx: f (x) g(x)dx + s f 2 a m a x) + The proof is complete. Remark 90 In Theorem 89, if we take m = 1, then f a+b 2 Z b g(x)dx a 1 2s 1 Z b f (x)g(x)dx; a which is an inequality of the Hermite-Hadamard-Féjer type for s-convex functions. Corollary 91 Under the same hypotheses of theorem 89, if g(x) = 1 and m = 1, we have: f 4.8 a+b 2 (b a) 1 2s 1 Z b f (x)dx: a For strongly (s; m) Convex Functions All of this result can be found in [118] Theorem 92 Let f : [0; +1) ! R be a strongly (s; m)-convex function, modulus c , where m 2 (0; 1] and let a; b 2 [0; +1) with a < b. Suppose 64 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY that f 2 L1 [a; b] and that g : [a; b] ! R is a nonnegative, integrable function a+b which is symmetric with respect to . Then 2 Zb f (b) + f (a) 2 f (x)g(x)dx a Zb x b s a a g(x)dx a b m a +f m 2 m f + Zb b b s x a g(x)dx a c 2 " a m b 2 b m + a 2 # Zb x b a b a b x g(x)dx: a a Proof. Let f and g as in the statement of the theorem. Then Zb f (x)g(x)dx a 2 b Zb Z 14 = f (x)g(x)dx + f (a + b 2 x)g(a + b a a 2 b Zb Z 14 f (x)g(x)dx + f (a + b = 2 1 = 2 Zb x)dx5 x)g(x)dx5 a a 3 3 (f (x) + f (a + b x))g(x)dx a 1 = 2 Zb f m x b a a b b xa x + am b a b +f a x b b a a+m a b x b am g(x)dx a 2 b Z 14 2 a s f (b) + m b b x a s f a m c x b a b a b x b a a m 2 65 4.8. FOR STRONGLY (S; M ) CONVEX FUNCTIONS + Zb s s x a x a b x b c f (a) + m f b a b a m b a a 2 b Z s s x a b x 14 f (f (b) + f (a)) + m = 2 b a b a a " a 2 x a b x b c + b a b a m f (b) + f (a) = 2 Zb x b b b x a a 2 3 b 5 g(x)dx m a b +f m m # # 2 b g(x)dx a m s a a g(x)dx a m f + a +f m 2 b m Zb b b s x a g(x)dx a c 2 " a m b 2 + a b m 2 # Zb x b a b a b x g(x)dx: a a Corollary 93 Let f : [0; +1) ! R be a strongly (s; 1)-convex function, modulus c , and let a; b 2 [0; +1) with a < b. Suppose that f 2 L1 [a; b] and that g : [a; b] ! R is a nonnegative, integrable function which is symmetric a+b . Then with respect to 2 Zb Zb Zb s s f (b) + f (a) x a b x f (x)g(x)dx g(x)dx c (x a)(b x)g(x)dx: + 2 b a b a a a a Theorem 94 Let f : [0; +1) ! R be a strongly (s; m)-convex function, modulus c , where s; m 2 (0; 1], and let a; b 2 [0; +1) with a < b. Suppose that f 2 L1 [a; b]; m 2 (0; 1] and that g : [a; b] ! R is a nonnegative, integrable a+b function which is symmetric with respect to . Then 2 Zb Zb Zb x x 2 a + b a+b x g(x)dx m f g(x)dx+2s 2 c 2s f g(x)dx 2 m m a a a 66 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY Zb f (x)g(x)dx: a Proof. In this case we have f a+b 2 Zb g(x)dx a = Zb a+b 2 f x + mx 2m g(x)dx a Zb 1 f (a + b 2s x) + c a+b 4 m x f s 2 m x m x 2 g(x)dx a 1 = s 2 Zb f (a + b m x)g(x)dx + s 2 a 1 = s 2 Zb = 1 2s x f g(x)dx m c 4 a f (a + b m x)dx + s 2 x)g(a + b a Zb Zb Zb a+b f (x)g(x)dx + m 2s a f x 2 g(x)dx m a Zb x f g(x)dx m c 4 x g(x)dx m Zb a+b c 4 a Zb a+b x x 2 g(x)dx; m a thus obtaining the required inequality. As a consequence of theorems 92 and 94 we get the following result. Corollary 95 Let f : [0; +1) ! R be a strongly (s; m)-convex function, modulus c , where s; m 2 (0; 1], and let a; b 2 [0; +1) with a < b. Suppose that f 2 L1 ([a; b]); m 2 (0; 1] and that g : [a; b] ! R is a nonnegative, a+b integrable function which is symmetric with respect to . Then 2 2s f a+b 2 Zb a g(x)dx m Zb a x a a Zb x x f g(x)dx2s 2 c m Zb a a+b x x 2 g(x)dx m x 2 g(x)dx m 4.9. FOR RELATIVE STRONGLY H CONVEX FUNCTIONS Zb f (x)g(x)dx f (b) + f (a) 2 Zb x b a a a +f m 2 b m a 67 s g(x)dx+ a m f + Zb b b x a # Zb x b s g(x)dx a c 2 4.9 " b a m 2 + a b m 2 a b a b x g(x)dx: a a For Relative Strongly h Convex Functions The following results treat a re…nement of the Hermite-Hadamard Inequality. For this goal we use the nest theorem. Theorem 96 Let h : (0; 1) ! (0; 1) be a given function. If a function f : I ! R is Lebesgue integrable and relative strongly h-convex with module c > 0, then Z g(b) 1 f (x)dx g(b) a a Z 1 c h(t)dt (f (a) + f (g(b))) (g(b) a)2 ; (4.17) 6 0 for all a; g(b) 2 I, a < g(b). Theorem 97 Let h : (0; 1) ! (0; 1) be a given function. If a function f : I ! R is Lebesgue integrable and relative strongly h-convex with module c > 0, then Z g(b) 1 f (x)dx g(b) a a Z 1 1 + 2h 12 h(t)dt (f (a) + f (g(b))) 2 0 Z 1 1 1 h(t)dt + c(g(b) a)2 ; (4.18) 4 0 24 68 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY for all a; g(b) 2 I, a < g(b). Applying the Theorem 96 in the intervals a; Proof. a + g(b) 2 and a + g(b) ; g(b) we obtain 2 Z 2 g(b) a a+g(b) 2 f (x)dx a f (a) + f a + g(b) 2 Z 1 h(t)dt 0 c (g(b) a)2 ; 6 4 (4.19) and 2 g(b) a f Z g(b) a+g(b) 2 a + g(b) 2 f (x)dx + f (g(b)) Z 1 h(t)dt 0 c (g(b) a)2 : 6 4 (4.20) Summing up these inequalities we get 2 g(b) a Z g(b) f (x)dx a a + g(b) 2 (f (a) + 2f + f (g(b))) Z 1 h(t)dt 0 2c (g(b) a)2 : 6 4 Therefore 1 g(b) a Z g(b) f (x)dx a f (a) + 2f a + g(b) 2 2 + f (g(b)) Z 1 h(t)dt 0 Now, using the relative strong h-convexity of f , we obtain c (g(b) a)2 : 6 4 69 4.9. FOR RELATIVE STRONGLY H CONVEX FUNCTIONS f h a + g(b) 2 1 f (a) + h 2 1 2 f (g(b)) c (a 4 g(b))2 : Thus, Z g(b) 1 f (x)dx g(b) a a Z 1 1 + 2h 21 h(t)dt (f (a) + f (g(b))) 2 0 Z 1 c (g(b) a)2 c 2 h(t)dt (a g(b)) 4 6 4 0 Z 1 1 1 + 2h 2 h(t)dt (f (a) + f (g(b))) = 2 0 Z 1 1 1 c(g(b) a)2 : h(t)dt + 4 0 24 Corollary 98 Under the same hypotheses of theorem ??, if h( 12 ) R1 h(t)dt 12 we get 0 1 2 and Z g(b) 1 f (x)dx g(b) a a Z 1 1 + 2h 21 h(t)dt (f (a) + f (g(b))) 2 0 Z 1 1 1 c(g(b) a)2 h(t)dt + 4 0 24 Z 1 c h(t)dt (f (a) + f (g(b))) (g(b) a)2 : 6 0 Corollary 99 If we take g(b) = b, then we get the right-hand side of the inequality given in [56]. 70 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY Remark 100 1. If we take c = 0 and h( 21 ) = 12 in the Theorem 4.18, then we have the right-hand side of the inequality given in [91, Theorem 16]. 2. If we take h(t) = ts with s 2 [0; 1] in Corollary 98, then we obtain Z 1 1 1 , 0 s 1; ts dt = s+1 2 0 and h 1 2 1 2 , , , 1 1 s 2 2 s 2 2 s 1 thus, the theorem is valid only for s = 1. 3. If we take h(t) = t for t 2 (0; 1) then the inequalities in the Corollary 98 reduce to Z g(b) (f (a) + f (g(b))) c 1 f (x)dx (g(b) a)2 ; g(b) a a 2 6 these is the hermite-Hadamard type inequalities for relative strongly convex functions. 4.10 For (m; h1; h2) Convex Functions In this section we establish some Ostrowski type inequalities for functions with (m; h1 ; h2 ) convex derivative functions. The following results have been compiled from the work of Vivas, M and García, C. (See [121]). First we have the following Lemma. Lemma 101 Let f : I R ! R a function di¤erentiable in int(I) where 0 a; b 2 I and a < b. If f 2 L[a; b], then Z b Z 1 (x a)2 1 0 f (x) f (u)du = tf (tx + (1 t)a)dt b a a b a 0 Z (b x)2 1 0 tf (tx + (1 t)b)dt: b a 0 for all x 2 [a; b]. 71 4.10. FOR (M; H1 ; H2 ) CONVEX FUNCTIONS Theorem 102 Let h1 ; h2 : [0; 1] ! R+ be non-negative functions and let f : I ! R a di¤erentiable function in int(I) and such that f 0 2 L [a; b] where a; b 2 I; a < b: Let m 2 (0; 1] : If jf 0 j is (m; h1 ; h2 ) convex function on I and jf 0 (x)j M for all x 2 [a; b] and for some M > 0, then f (x) 1 b a Z b M f (u)du b a a 2 (x a) + (b x) 2 Z 1 t (h1 (t) + mh2 (t)) dt 0 holds for all x 2 [a; b] : Proof. Using Lemma (101) and de…nition (39) we have: f (x) 1 b a (x b Z b f (u)du a Z a)2 a 1 tjf 0 (tx + (1 0 (b + b (x = b a)2 a Z 1 a)2 a 0 Z Z 1 tjf 0 (tx + (1 1 0 x)2 a t)b)jdt 0 tjf 0 tx + m(1 (b + b (x b x)2 a t)a)jdt Z t) a jdt m 1 tjf 0 tx + m(1 0 t h1 (t)jf 0 (x)j + mh2 (t)jf 0 t) b m jdt a j dt m Z x)2 1 b t h1 (t)jf 0 (x)j + mh2 (t)jf 0 j dt a 0 m Z (x a)2 1 t(h1 (t) + mh2 (t))dt M b a 0 Z (b x)2 1 +M t(h1 (t) + mh2 (t))dt b a 0 Z 1 M 2 2 t(h1 (t) + mh2 (t))dt; ((x a) + (b x) ) = b a 0 (b + b which proves the result. 72 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY 4.11 For (m; h1; h2) GA Convex Functions In Appendix A we …nd some basic knowledge about Stochastic Processes. The following results can be found in the Vivas,M. and Hernández,J. publication [117]. De…nition 103 Let ( ; A; P ), be a, probability space T R be an interval, we say that a stochastic processes X : [0; b) ! [0; +1) is (h1 ; h2 ; m)-GA convex if X(u v (1 )m( ) ; ) h1 ( )X(u; ) + m( )h2 (1 )X(v; ) for all u; v 2 [0; 1], with hi : [0; 1] ! R0 and m : [0; 1] ! (0; 1] such that hi 6 0 for i=1,2. Theorem 104 Let hi : [0; 1] ! R0 , where hi 6 0 for i = 1; 2; m : [0; 1] ! (0; 1], and X : [0; +1) ! R0 be an (h1 ; h2 ; m)-GA-convex function on and hi 2 L1 [a; b] for 0 < a < b. (0; m(b1 ) ] 2 Then Z b Z b p m( 12 )h2 ( 21 ) h1 ( 12 ) t X( ab; ) X(t; )dt + X( 1 ; )dt ln(b) ln(a) a ln(b) ln(a) a m( 2 ) Proof. Since p 1 1 ab = (at :b1 t ) 2 :(a1 t :bt ) 2 for 0 t i1, from the (h1 ; h2 ; m)-GA-convexity of the stochastic process X , we obtain on 0; m(b1 ) 2 p X( ab; ) h1 1 2 1 2 X(at b1 t ; ) + m h2 1 2 X a1 t b t ; m( 12 ) Integrating both sides of the above inequality and replacing the argument, in the right side, a1 t :bt and at b1 t for 0 t 1 by s, then Z b Z 1 1 1 t t X(s; ) (4.21) X(a b ; )dt = ln(b) ln(a) a 0 and Z 0 1 X a1 t b t ; m( 21 ) dt = 1 ln(b) The proof of theorem is complete. ln(a) Z a b X s ; m( 12 ) ds (4.22) 4.11. FOR (M; H1 ; H2 ) GA CONVEX FUNCTIONS 73 Theorem 105 Let hi : [0; 1] ! R0 , where hi 6 0 for i = 1; 2; m : [0; 1] ! (0; 1], and X : [0; +1) ! R0 be an (h1 ; h2 ; m)-GA-convex stochastic b process on (0; m ] such that X is an integrable stochastic process in [a; mb ] and h1 ; h2 2 L1 ([0; 1]) for 0 < a < b, then Z b 1 X(t; )dt minfA; Bg; ln(b) ln(a) a where A = X(a; ) Z b ; m Z a ; h1 (t)dt + mX m Z 1 h1 (t)dt + mX 0 and B = X (b; ) Z 1 0 1 h2 (t)dt 0 1 h2 (t)dt: 0 If h1 (t) = h2 (t) = h(t) for all t 2 [0; 1], we have Z b 1 X(t; )dt minfC; Dg; ln(b) ln(a) a where C= and X(a; ) + mX b ; m a D = X (b; ) + mX ; m Z 1 h(t)dt 0 Z 1 h(t)dt: 0 Proof. Letting x = a1 t bt for 0 t 1, by the (h1 ; h2 ; m)-GA-convexity of X and (4.21), we obtain Z 1 Z b 1 X(a1 t bt ; )dt minfA; Bg; X(t; )dt = ln(b) ln(a) a 0 where A = X(a; ) Z B = X(b; ) Z 0 The proof is complete. Z a h1 (t)dt + mX ; m Z h1 (t)dt + mX 0 and b ; m 1 1 1 h2 (t)dt 0 0 1 h2 (t)dt 74 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY Theorem 106 Let hi : [0; 1] ! R0 ,hi 6 0 for i = 1; 2; m 2 (0; 1]; X : [0; +1) ! R0 be an (h1 ; h2 ; m)-GA-stochastic process on 0; mb2 such that X is integrable in [a; mb ] and h1 ; h2 2 L1 ([0; 1]) for 0 < a < b then p X( ab; ) Z b Z b h2 21 h1 ( 21 ) t ; X(t; )dt + m X ln(a) ln(b) a ln(b) ln(a) a m Z 1 Z 1 h2 (t)dt ; h1 (t)dt + mB min A 0 0 Z 1 Z 1 C h2 (t)dt h1 (t)dt + mD 0 0 where 1 2 A = h1 1 2 B = h1 D = h1 b ; m 1 2 1 2 1 2 X a ; m + mh2 1 2 X b ; m2 X (b; ) + mh2 1 2 X b ; m 1 2 X a ; m2 X C = h1 and X(a; ) + mh2 X a ; m + mh2 ; ; : Proof. From the (h1 ; h2 ; m)-GA-convexity of X on 0; mb2 , we obtain X p ab; h1 1 2 X at b 1 t ; 1 2 min h1 + mh2 h1 1 2 +mh2 + mh2 1 2 X h1 (t)X (a; ) + mh2 (1 a1 t b t ; m t)X b ; m a b 1 h1 (1 t)X( ; ) + mh2 (t)X( 2 ; ) ; 2 m m h i a h1 (1 t)X (b; ) + mh2 (t)X ; m 1 a b ; h1 (t)X ; + mh2 (1 t)X 2 m m2 dt 4.11. FOR (M; H1 ; H2 ) GA CONVEX FUNCTIONS 75 Substituting a1 t bt and at b1 t for 0 t 1 by u an integrating on both sides of the above inequality with respect to t 2 [0; 1] lead to Z b Z b p h2 21 h1 21 u X X (u; ) du + m X ; du ab; ln(b) ln(a) a ln(b) ln(a) a m b 1 h1 (t)X(a; ) + mh2 (1 t)X ; min h1 2 m a b 1 h1 (1 t)X( ; ) + mh2 (t)X ; ; + mh2 2 m m2 i 1 h a h1 h1 (1 t)X(b; ) + mh2 (t)X ; 2 m a 1 b h1 (t)X ; + mh2 (1 t)X ; +mh2 2 m m2 The theorem is proved. Theorem 107 Let hi : [0; 1] ! R0 ,hi 6 0 for i = 1; 2; and h1 (t1 )h2 (t2 ) h2 (t1 t2 ) for all t1 ; t2 2 [0; 1] and h2 be a supermultiplicative function. Let m : [0; 1] ! (0; 1] and X : [0; +1) ! R0 be a (h1 ; h2 ; m) GA convex stochastic process. Then ! n Q 1 Y m(wj ) wi ij=0 X ti ; (4.23) i=1 h1 (w1 )X (t1 ; ) + i 1 n Y X i=2 holds for all ti 2 (0; b]; wi > 0 with j=1 Pn i=1 ! m(wj ) h2 (wi )X(ti ; ) wi = 1 and m(w0 ) = 1 Proof. Using induction over n . When n = 2 taking t = w1 and 1 t = w2 in De…nition 103 we obtain 4.23. Suppose that for n = k the inequality 4.23 holds, that is ! k Qi 1 Y m(wj ) wi ; (4.24) X ti j=0 i=1 h1 (w1 )X (t1 ; ) + i 1 k Y X i=2 j=1 ! m(wj ) h2 (wi )X(ti ; ) 76 CHAPTER 4. INEQUALITIES AND GENERALIZED CONVEXITY When n = k + 1, letting Sk = hypothesis we have 1 = X @tw 1 i=1 n Y X 0 Pk+1 wi , by De…nition 103 and the induction wi ti Qi 1 j=0 m(wj ) ; i=1 k+1 Y wi =Sk ti Qi 1 j=0 m(wj ) i=2 ! !m(w1 )Sk 1 ; A h1 (w1 )X (t1 ; ) w =Sk t2 2 + m(w1 )h2 (Sk )X k+1 Y wi =Sk ti Qi 1 j=2 m(wj ) ; i=3 ! h1 (w1 )X (t1 ; ) + m(w1 )h2 (Sk ) h1 + k+1 i 1 X Y i=3 j=2 w2 Sk X(t2 ; ) ! m(wj ) h2 wi Sk X(ti ; ) # Since h2 is a supermultiplicative function, we have h2 (Sk )h2 (wi =Sk ) h2 (wi ) for i = 1; 2; :::; n. This implies that when n = k + 1 the inequality 4.23 holds. The proof is complete. Chapter 5 Open Problems With the intention of stimulating research in the area developed in this brief compilation of results about generalized convexity, we propose some open problems below. 5.1 Problem #1 Some classic inequalities such as Chebyshev and Gruss can be established with some of the generalized convexity types mentioned in Chapter 3. For two absolutely continuous functions f ; g : [a; b] ! R; consider the functional Z b Z b Z b 1 1 1 T (f; g) = f (x)g(x)dx f (x)dx g(x)dx b a a b a a b a a µ provided, the involved integrals exist. In 1882, P. L. Cebyšev (see [7]) proved 0 0 that, if f ; g 2 L1 [a; b] then, T (f; g) 1 (b 12 a)2 kf 0 k1 kg 0 k1 ; and in 1934, G. Grüs (see [7]) showed that T (f; g) 1 (M 4 m) (N n); provided m; M; n and N are real numbers satisfying the conditions, 1<m f (x) 77 M <1 78 CHAPTER 5. OPEN PROBLEMS 1<n g(x) N < 1; for all x 2 [a; b] : Let’s see some results about these inequalities. First, Ra…q y Ahmad in [99] established a Grüss type inequality for functions whose second derivatives, in absolute value, are convex. Let f; g : [a; b] ! R be absolutely continuous on [a; b]. If jf 00 j and jg 00 j are convex on [a; b] and f 00 ; g 00 2 Lp [a; b], then, T (f; g) 1 12 (b a)2 Z bh 2 jb a where aj1=p (jf (x)j jg 00 (x)j + jg(x)j jf 00 (x)j) i + jf (x)j kg 00 kp + jg(x)j kf 00 kp Q(x)dx Q(x) = " (b x)2q+1 + (x 2q + 1 a)2q+1 #1=q and 1=p + 1=q = 1: The next result establishes an inequality of Chebyshev type (See [99]). Theorem 108 Let f; g : [a; b] ! R be absolutely continuous on [a; b]. If jf 00 j and jg 00 j are convex on [a; b] and f 00 ; g 00 2 Lp [a; b], then, Te(f; g) for all x 2 [a; b] : 1 36 (b a)3 Z b nh i jg(x)j 2 jf 00 (x)j (b a)1=p + kf 00 kp a h io 1=p 00 00 jf (x)j 2 jg (x)j (b a) + kg kp Q2 (x)dx In these two theorems we can observe the condition of convexity imposed on the second derivative of the functions involved. These results can be generalized by requesting that the second derivatives be from one of the types of generalized convexity discussed in Chapter 3. In addition to this, you can search for applications of them. 79 5.2. PROBLEM #2 5.2 Problem #2 In [38], Erdem, Ogumez and Gudat, took as a base the following result. Theorem 109 Let h : (0; 1) ! (0; 1) be a given function. If a function f :I R ! R is Lebesgue integrable and strongly h convex with modulus c > 0, then 1 f h(1=2) c + (b 12 a+b 2 a) 1 2 Z b f (x)dx Z 1 h(t)dt (f (a) + f (b)) b a a 0 c (b 6 a)2 : From this Theorem they took advantage of the strongly h convexity and they proved the following result demanding the condition of strongly s convexity for the second derivative, in modulus. Their result is as follow. Theorem 110 Suppose that f : I [0; 1] ! R be a twice di¤erentiable mapping on int(I) such that f 00 2 L([a; b]), where a; b 2 I with a < b. If jf 00 j is strongly s convex on [a; b], for some s 2 (0; 1] with modulus c > 0 , then the following inequalities hold f a+b 2 1 b a Z b f (x)dx a (b a)2 8(s + 1)(s + 2)(s + 3) jf 00 (a)j + (s + 1) (s + 2) f 00 c (b 160 (b a)2 8(s + 1)(s + 2)(s + 3) 1 + (s + 2) 21 [1 + (s + 1) (s + 2) 21 s ] c (b 12 a+b 2 + jf 00 (b)j a)2 s [jf 00 (a)j + jf 00 (b)j] a)2 c (b 160 a)2 : Changing the convexity condition by some of the generalized convexity concepts , it can be found many results of that type. 80 CHAPTER 5. OPEN PROBLEMS 5.3 Problem #3 Also in [38] , the same authors obtain results that demand the same condition of generalized convexity for the qth power of the second derivative in modulus. Theorem 111 Suppose that f : I [0; 1] ! R be a twice di¤erentiable 00 mapping on int(I) such that f 2 L([a; b]), where a; b 2 I with a < b. If jf 00 jq is strongly s convex on [a; b], for some s 2 (0; 1] with modulus c > 0 , then the following inequalities hold Z b 1 a+b f f (x)dx 2 b a a (b a)2 16 " 1=p 1 2p + 1 1 (s + 1) f 00 a+b 2 1 + (s + 1) f 00 q q 00 + jf (a)j a+b 2 c (b 6 q q + jf 00 (b)j 1=q a) 2 c (b 6 1=q a)2 # : Using other types of generalized convexity, results can be established that are no less important than those mentioned above. 5.4 Problem #4. In [76], Meftah, B. in 2017, proved a Ostrowski type inequality for functions whose n th derivative is ' convex in modulus. Theorem 112 Let f : I ! R be n-times di¤erentiable on [a; b] such that f (n) 2 L [a; b]. If f (n) is '-convex, then the following inequality " # Z b n X (b x)k+1 + ( 1)k (x a)k (k) f (x)dx f (x) (k + 1)! a k=0 (x a)n+1 n! 1 1 f (n) (a) + ' f (n) (a) ; f (n) (x) n+1 n+2 81 5.5. PROBLEM #5 + (b x)n+1 n! 1 1 f (n) (b) + ' f (n) (x) ; f (n) (b) n+1 (n + 1) (n + 2) holds for all x 2 [a; b] : It is proposed to make a similar result for functions whose n-th derivative in modulus is s-convex, (s; m) convex, strongly (s; m) convex. Remark 113 All the proposed problems can be studied in environments such as those of the real line in fractal sets, stochastic processes, self-adjoint operators in Hilbert spaces. 5.5 Problem #5 In the work of Roman Ger [44], we …nd the main subject of his study. He deal with the following two functional inequalities Z y Z y 1 x+y x+y f (t)dt g(t)dt g f 2 y x x 2 x and f (x) f (y) 2 Z y f (t)dt x 1 y x Z x y g(t)dt g(x) + g(y) 2 for functions f and g mapping an open interval I of the real line R into a Banach space and into R, respectively. If we take a close look of this inequalities, one can see that he treat a Hermite-Hadamard inequality type. He propose, at the end of the article, make the same study with other type of the classical arithmetic mean M= a+b : 2 We say that it would also be interesting to do the study with another class of inequalities or another type of generalized convexity in the environment of normed linear spaces. 82 CHAPTER 5. OPEN PROBLEMS Appendix A About Self-adjoint Operators in Hilbert Spaces The theory of operator/matrix monotone functions was initiated by the celebrated paper of C. Löwner [73], which was soon followed by F. Kraus [69] on operator/matrix convex functions. After further developments due to some authors (for instance, J. Bendat and S. Sherman [13], A. Korányi [67], and U. Franz [43]), in their seminal paper [52] F.Hansen and G.K. Pedersen established a modern treatment of operator monotone and convex functions. In [3, 15, 29, 54] are found comprehensive expositions on the subject matter. In order to achieve our results we need the following de…nitions and preliminary. With B(H) we shall denote the C algebra commutative of all bounded operators over a Hilbert space H with inner product h; i : Let A be a subalgebra of B(H): An operator A 2 A is positive if hAx; xi 0 for all x 2 H: Over A there exists an order relation by means A B if hAx; xi hBx; xi B A if hBx; xi hAx; xi or for A; B 2 A self-adjoint operators and for all x 2 H: The Gelfand map established a isometrically isomorphism between the set C( (A)) of all continuous functions de…ned over the spectrum of A, denoted by (A), and the C algebra C (A) generated by A and the identity operator 1H over H as follows: For any f; g 2 C( (A)) and ; 2 C (Complex numbers) we have 83 84APPENDIX A. ABOUT SELF-ADJOINT OPERATORS IN HILBERT SPACES 1. ( f + g) = 2. (f g) = 3. (A) + (B) (A) (B) and f = (f ) k (f )k = kf k := sup jf (t)j t2 (A) 4. (f0 ) = 1H and t 2 (A) (f1 ) = A; where f0 (t) = 1 y f1 (t) = t for all with this notation we de…ne f (A) = (f ) and we call it the continuous functional calculus for a self-adjoint operator A: If A is a self-adjoint operator and f is a continuous real valued function on (A) then f (t) 0 for all t 2 (A) ) f (A) 0 that is to say f (A) is a positive operator over H: Moreover, if both functions f; g are continuous real valued functions on (A) then f (t) g(t) for all t 2 (A) ) f (A) g(A) respect to the order in B(H): De…nition 114 Let H be a Hilbert space and I R an interval. A continuous function f : I ! R is called operator convex with respect to H if f ( A + (1 ) B) f (A) + (1 ) f (B) for all A; B 2 B(H)sa with (A) [ (B) I and for all scalars 2 [0; 1]. f is called operator convex of order n 2 N if it is operator convex with respect to H = C n : Finally, f is simply called operator convex if there is an in…nite dimensional Hilbert space H such that f is operator convex with respect to H: Here B(H)sa is the set of self-adjoint bounded operators on the Hilbert space H, (A); (B), denotes the spectrum of A and B, and f (A) and f (B) are de…ned by the continuous functional calculus. We refer the reader to [83] for unde…ned notions on C algebra theory. As illustration below we state some classical theorems on operator inequalities. 85 Theorem 115 [Bendat and Sherman [13]] f is operator convex if and only if it is operator convex of every order n 2 N , and this last property holds if and only if it is operator convex with respect to the Hilbert space `2 (C). Theorem 116 [F. Hansen and G.K. Pedersen [52]] A continuous function f de…ned on an interval I is operator convex if and only if ! X X f aj f (xj )aj aj x j aj j2J j2J for every …nite family fxj : j 2 Jg of bounded, self-adjoint operators on a separable Hilbert space H, with spectra P contained in I, and every family of operators faj : j 2 Jg in B(H) with j2J aj aj = 1; where 1 2 B(H) is the identity operator. Theorem 117 [D.R. Farenick and F. Zhou [41]] Let ( ; ; ) be a probability measure space, and suppose f is an operator convex function de…ned on an open interval I R: If g : ! B(C n )sa is a measurable function for which (g(!)) [ ; ] I for all ! 2 , then Z Z f gd : gd f 86APPENDIX A. ABOUT SELF-ADJOINT OPERATORS IN HILBERT SPACES Appendix B About Convex Stochastic Processes The study on convex stochastic processes began in 1974 when B. Nagy in [84], applied a characterization of measurable stochastic processes to solving a generalization of the (additive) Cauchy functional equation. In 1980, Nikodem [87] introduced the convex stochastic processes in his article. Later in 1995, A. Skrowronski in [111] presented some further results on convex stochastic processes. In 2014 Maden et. al. [74] introduced the convex stochastic processes in the …rst sense and proved Hermite-Hadamard type inequalities to these processes. In the year 2014, E. Set et. al. in [106] investigated Hermite-Hadamard type inequalities for stochastic processes in the second sense. They investigated a relation between s-convex stochastic processes in the second sense and convex stochastic processes. For other results related to stochastic processes see [8, 9, 27, 78] where further references are given. De…nition 118 Let ( ; F; P ) be an arbitrary probability space. A function X : ! R is called a random variable if it is F-measurable. Let ( ; F; P ) be an arbitrary probability space and let T R be time. A collection of random variable X(t; w); t 2 T with values in R is called a stochastic processes. 1. If X(t; w) takes values in S = Rd if is called vector-valued stochastic process. 87 88 APPENDIX B. ABOUT CONVEX STOCHASTIC PROCESSES 2. If the time T can be a discrete subset of R, then X(t; w) is called a discrete time stochastic process. 3. If the time T is an interval, R+ or R, it is called a stochastic process with continuous time Throughout the book we restrict our attention stochastic process with continuous time, i.e, index set T = [0; +1). De…nition 119 Set ( ; A; P ) be a probability space and T val. We say that a stochastic process X : T ! R if R be an inter- 1. Convex if X( u + (1 )v; ) X(u; ) + (1 )X(v; ) (B.1) t)X(v; ) (B.2) for all u; v 2 T and 2 [0; 1]. This class of stochastic process are denoted by C. 2. m-convex if X(tu + m(1 t)v; ) tX(u; ) + m(1 for all u; v 2 T and t 2 [0; 1]; m 2 (0; 1]. De…nition 120 Let ( ; A; P ) be a probability space and T val. We say that the stochastic process X : ! R is called R be an inter- 1. Continuous in probability in interval T if for all t0 2 T P limt!t0 (t; ) = X(t0 ; ) where P lim denotes the limit in probability; 2. mean-square continuous in the interval T if for all t0 2 T P limt!t0 E(X(t; ) X(t0 ; )) = 0 where E(X(t; )) denotes the expectation value of the random variable X(t; ); 3. increasing (decreasing) if for all u; v 2 T such that t < s, X(u; ) X(v; ); (X(u; ) X(v; )) 89 4. monotonic if it’s increasing or decreasing; 5. di¤erentiable at a point t 2 T if there is a random variable X 0 (t; ) : T ! RX 0 (t; ) = P lim t!t0 X(t; ) t X(t0 ; ) t0 We say that a stochastic process X : T ! R is continuous (di¤erentiable) if it is continuous (di¤erentiable) at every point of the interval T . [68, 110, 111, 87] De…nition 121 Let ( ; A; P ) be a probability space T R be an interval 2 with E(X(t) ) < 1 for all t 2 T . Let [a; b] T; a = t0 < t1 < ::: < tn = b be a partition of [a; b] and k 2 [tk 1 ; tk ] for k = 1; 2; :::; n. A random variable Y : ! R is called mean-square integral of the process X(t; ) on [a; b] if the following identity holds: lim E[X( k (tk n!1 tk 1 ) Y )2 ] = 0 Then we can write Z b X(t; )dt = Y ( )(a:e:) a Also, mean square integral operator is increasing, that is, Z b Z b Z(t; )dt(a:e:) X(t; )dt a a Where X(t; ) Z(t; ) in [a; b] [108] In throughout paper, we will consider the stochastic processes that is with continuous time and mean-square continuous. 90 APPENDIX B. ABOUT CONVEX STOCHASTIC PROCESSES Appendix C About Real Line on Fractal Sets Fractals have been known for about more than a century and have been observed in di¤erent branches of science. But it is only recently (approximately in the last forty years) that they have become a subject of mathematical study. The pioneer of the theory of fractals was Benoit Mandelbrot. His book Fractals: Form, Chance and Dimension …rst appeared in 1977, and a second, enlarged, edition was published in 1982. Since that time, serious articles, surveys, popular papers, and books about fractals have appeared by the dozen. Mandelbrot in [75] de…ned a fractal set is one whose Hausdor¤ dimension exceeds strictly its topological dimension. Also, Yang in [134] established the numerical sets, where is the dimension of the considered fractal. For more details about fractal sets see for instance [36, 39, 40, 134] and references therein. Recently, the theory of Yang’s fractional set of elements sets was introduced as follows: For 0 < 1 we have the following type sets. Z = f0 ; 1 ; 2 ; :::; n ; :::g Q = f(a=b) : a 2 Z ; b 2 Z ; b 6= 0 g I = fm 6= (a=b) : a 2 Z ; b 2 Z ; b 6= 0 g R =Q [I For a ; b ; c 2 R the following properties hold : 91 92 APPENDIX C. ABOUT REAL LINE ON FRACTAL SETS a. a + b 2 R y a b 2 R b. a + b = b + a = (a + b) = (b + a) c. a + (b + c ) = (a + b ) + c d. a b = b a = (ab) = (ba) e. a (b c ) = (a b ) c f. a + 0 = 0 + a = a y a 1 = 1 a = a If a b is non negative we say a is greater than or equal to b , or b is less than or equal to a ; and we write a b or b a , respectively. If there is not possibility that a = b then we write a > b o b < a . Next we recall some de…nitions and some facts of fractional calculus theory on R which will be used in this paper. De…nition 122 Let f : R ! R be a mapping. We say that f is local fractional continuous at x0 2 R; if for all > 0 exists > 0 such that jx x0 j < =) jf (x) f (x0 ) j < If f is local fractional continuous in each point of an interval (a; b), we say that f is local fractional continuous in (a; b) and we write f 2 C (a; b) : De…nition 123 The local fractional derivative of f of order de…ned by f ( ) (x0 ) = where (f (x) d f (x) dx f (x0 )) = = lim x=x0 x!x0 ( + 1) (f (x) at x = x0 is (f (x) f (x0 )) (x x0 ) f (x0 )) : De…nition 124 Let f 2 C [a; b]. Then the local fractional integral of order of f is de…ned by Z b 1 ( ) f (x) (dx) a Ib f = (1 + ) a N X 1 = lim f (ti ) ( ti ) (1 + ) t!0 i=1 93 where ti = ti+1 ti , t = maxf t1 ; :::; tN g, and [ti ; ti+1 ], i = 1; 2; :::; N , with a = t0 < t1 < ::: < tN 1 = b; is a partition of [a; b]. ( ) ( ) If for each x 2 [a; b] there exists a Ib f; then we write f 2 Ix [a; b]: Here, it follows ( ) a Ib f and ( ) b Ia f = = 0 if a = b ( ) a Ib f if a < b: Also we have the property of change of variables. 94 APPENDIX C. ABOUT REAL LINE ON FRACTAL SETS Afterword We hope that this minibook serves to stimulate research in this area of Mathematics. Readers are invited to review the exposed bibliography in such a way that , with the recent results shown in this one, can have an appropriate address for future publications. 95 96 AFTERWORD Bibliography [1] M. Alomari , , M. Darus , S.S. Dragomir, P. Cerone. Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense. Applied Mathematics Letters 23 (2010) 1071 1076 [2] T. Ando, F. Hiai. Operator log-Convex Functions and Operator Means. arXiv: 0911.5267v5, 2014 [3] T. Ando, Topics on Operator Inequalities . (mimeographed). Hokkaido Univ. Sapporo, 1978. Lecture Notes [4] J.L. Aujla, H. L. Vasudeva. 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