Classical Analysis

When Newton's method, or Halley's method is used to approximate the pth root of 1−z, a sequence of rational functions is obtained. In this paper, a beautiful formula for these rational functions is proved in the square root case, using an interesting link to Chebyshev's polynomials. It allows the determination of the sign of the coefficients of the power series expansion of these rational functions. This answers positively the square root case of a proposed conjecture by Guo .

The critical buckling load and the natural frequency of frame structures are usually determined to avoid failures due to instability and resonance. Classical analyses are intractable and many analysts resort to numerical method of analysis using the traditional 4x4 matrix stiffness systems, which also have great limitation. This work presents new 5x5 stiffness matrices for classical and effective stability and dynamic analyses of line continua. Energy variational principle was employed in developing the matrices, with five term Mclaurin's polynomial series as the shape function. A central deflection node was considered, making a total of five deformable nodes. The new 5x5 matrices were derived by minimizing the geometric work and kinetic energy of the line continuum of five term shape function. They were employed, as well as the traditional 4x4 matrices, in classical stability and free vibration analyses of four line continua and a portal frame. The results from the new 5x5 matrices were very close to exact results, with average percentage differences of 2.55% for stability and 0.14% for free vibration, whereas those from the traditional 4x4 matrices differed greatly from exact results, with average percentage differences of 23.73% for stability and 14.72% for free vibration. Thus, the newly developed stiffness matrices are suitable for stability and dynamic analyses of line continua and should be used by structural engineering analysts accordingly.

This purpose of this paper is to note an interesting identity derived from an integral in Gradshteyn and Ryzhik using techniques from George Boros'(deceased) Ph.D thesis. The idenity equates a sum to a product by evaluating an integral in two different ways. A more general form of the idenity is left for further investigation.

Statistical convergence and statistical Cauchy sequence in 2-normed space were studied by Gürdal and Pehlivan [M. Gürdal, S. Pehlivan, Statistical convergence in 2-normed spaces, Southeast Asian Bulletin of Mathematics, (33) (2009), 257-264]. In this paper, we get analogous results of statistical convergence and statistical Cauchy sequence of functions and investigate some properties and relationships between them in 2-normed spaces.

In this note we prove optimal inequalities for bounded functions in terms of their deviation from their mean. These results extend and generalize some known inequalities due to Thong (2011) and Noting that for continuous functions f from F m,M we have 2010 Mathematics Subject Classification. 26B20, 26D25.

Orthogonal polynomials have very useful properties in the mathematical problems, so recent years have seen a great deal in the  field of approximation theory using orthogonal polynomials. In this paper, we characterize a sequence of the generalized Chebyshev-type polynomials of the first kind , which are orthogonal with respect to the measure.
Then we provide a closed form of the constructed polynomials in term of the Bernstein polynomials.
We conclude the paper with some results on the integration of the weighted generalized Chebyshev-type with the Bernstein polynomials.

In this note, we recall Kummer's Fourier series expansion of the 1-periodic function that coincides with the logarithm of the Gamma function on the unit interval (0, 1), and we use it to find closed forms for some numerical series related to the generalized Stieltjes constants, and some integrals involving the function x → ln ln(1/x).

We establish necessary and sufficient conditions for an arbitrary polynomial of degree n, especially with only real roots, to be trivial, i.e. to have the form a(x − b) n. To do this, we derive new properties of polynomials and their roots. In particular, it concerns new bounds and genetic sum representations of the Abel-Goncharov interpolation polynomials. Moreover, we prove the Sz.-Nagy type identities, the Laguerre and Obreshkov-Chebotarev type inequalities for roots of polynomials and their derivatives. As applications these results are associated with the known problem, conjectured by Casas-Alvero in 2001, which says, that any complex univariate polynomial, having a common root with each of its non-constant derivative must be a power of a linear polynomial. We investigate particular cases of the problem, when the conjecture holds true or, possibly, is false.

In this note we deal with some inequalities for the tangent function that are valid for x in (−π/2, π/2). These inequalities are optimal in the sense that the best values of the exponents involved are obtained. 2010 Mathematics Subject Classification. 26D05. Key words and phrases. inequality, tangent function, power series expansion.

Hilbert transform of wavelets has been used to approximate functions in L 2 (R). It is proved that Hilbert transform of wavelets with many vanishing moments does a good job in approximating smooth functions in L 2 (R). We also prove that Hölder continuity of a function helps in the decay of wavelet coefficients and thereby helps in approximating it. Finally, we give a result that relates the Hilbert transform of wavelet with dyadic scale differential operator and use it to decrease the wavelet coefficients.

In this paper, we develop a generalization of q-Bernstein-Kantorovich type operators. We first study some fundamental properties of these operators and then investigate approximation properties of a sequence of these operators using Korovkin theorem. Finally, we estimate rate of approximation by modulus of continuity.

We construct multiple representations relative to different bases of the generalized Tschebyscheff polynomials of second kind. We build the change-of-basis matrices between the generalized Tschebyscheff of the second kind polynomial basis and Bernstein polynomial basis. Also, we provide an explicit closed from of The generalized Polynomials of degree r less than or equal n in terms of the Bernstein basis of fixed degree n.

In this paper, we present a family of multivariable polynomials defined by Rodrigues formula and we discuss their some miscellaneous properties such as generating function and recurrence relation. We also derive various classes of multilateral generating functions for these multivariable polynomials and give some special cases of the results. Furthermore, we also show that some particular cases of the polynomials reduce to the products of Hermite and Laguerre orthogonal polynomials with one variable.

For a function f which is analytic and univalent in the unit disk {z ∈ C : |z| < 1} having the power series expansion of the normalized form z + ∑ ∞ n=2 a n z n , Zalcman conjectured that |a 2 n − a 2n−1 | (n − 1) 2 , n = 2,3,.... In this article, we obtain the sharp estimate for the classical Zalcman coefficient functional a 2 n − a 2n−1 for the above class of functions with the restriction that the n-th coefficient, a n , has certain integral representation associated with probability measure. Moreover, we also study a similar problem for the classes of functions of the above form whose coefficients satisfy certain inequalities.

Silverman [4] was defined the class of univalent functions f (z) = z + ∞ ∑ k=2 a k z k for which arg(a k) prescribed in such way that f (z) is univalent if and only if f (z) is starlike. In this paper we introduce the subclass of p-valent functions with varying arguments, especially p-valent starlike functions and p− valent convex functions, moreover we give some interesting properties of functions in these classes, including coefficients estimates, distortion theorems and extreme functions.

Given two positive real numbers x and y, let A(x,y), G(x,y), and I(x,y) denote their arithmetic mean, geometric mean, and identric mean, respectively. Also, let Kp(x,y) = p q 2 3 A p (x,y) + 1 3 G p (x,y) for p &gt; 0. In this note we prove that Kp(x,y) &lt; I(x,y) for all positive real numbers x6 y if and only if p 6/5, and that I(x,y) &lt; Kp(x,y) for all positive real numbers x6 y if and only if p (ln3 ln2)/(1 ln2). These results, complement and extend similar inequalities due to J. Sandor (2), J. Sandor and T. Trif (3), and H. Alzer and S.-L. Qiu (1).

In this paper, the approximation for the class of functions L 1 (Γ) is investigated by means of rational functions of the form R n (z) = ∑ n k=−n a k (z−b) k. This class is difficult of access and little studied. The functions from L 1 (Γ) satisfying natural condition of Lipschitz on the curve Γ , namely, f (z(s + h)) − f (z(s)) L 1 (Γ) const|h| α are considered. The corresponding approximation theorem is proved.

In the present paper we introduce and studied two subclasses of multivalent functions denoted by M λ p,n (γ;β) and N λ p,n (μ,η;δ). Further, by giving specific values of the parameters of our main results, we will find some connection between these two classes, and moreover, several consequences of main results are also discussed.

In the present paper, we consider the Bézier variant of the general family of Gupta-Srivastava operators [7]. For the proposed operators, we discuss the rate of convergence by using of Lipschitz type space, Ditzian-Totik modulus of smoothness, weighted modulus of continuity and functions of bounded variation.

In the present paper we study a Phillips type modified Bernstein operator M n , where the function is defined in the mobile interval 0,1 − 1 n+1 and obtain its m − th order moment. We establish some direct results in simultaneous approximation for this modified Bernstein operator. Mathematics subject classification (2010): 41A25, 41A36.

In this paper, we construct generalized Tschebyscheff-type weighted orthogonal polynomials in the Bernstein-Bezer form over the simplicial domain. We show that ..., form an orthogonal system over a triangular domain with respect to the generalized weight function