Sequences of some meromorphic function spaces

Sequences of some meromorphic function spaces A. El-Sayed Ahmed M. A. Bakhit Abstract Our goal in this paper is to introduce some new sequences of some meromorphic function spaces, which will be called bq and qK -sequences. Our study is motivated by the theories of normal, Q#K and meromorphic Besov functions. For a non-normal function f the sequences of points {an } and {bn } for which lim (1 − |an |2 ) f # ( an ) = +∞ and n→∞ lim or n→∞ ZZ ∆
q f # (z) (1 − |z|2 )q−2 (1 − | ϕ an (z)|2 )s dA(z) = +∞ lim n→∞ ZZ ∆
2 f # (z) K (z, an )dA(z) = +∞ are considered and compared with each other. Finally, non-normal meromorphic functions are described in terms of the distribution of the values of these meromorphic functions. 1 Introduction Let ∆ = {z : |z| < 1} be the open unit disk in the complex plane C and let dA(z) be the Euclidean area element on ∆. Let M(∆) denote the class of functions meromorphic in ∆. The pseudohyperbolic distance between z and a is given by z is the Möbius transformation of ∆. For σ(z, a) = | ϕ a (z)|, where ϕ a (z) = 1a−−āz 0 < r < 1, let ∆(a, r ) = {z ∈ ∆ : σ(z, a) < r } be the pseudohyperbolic disk with Received by the editors February 2008. Communicated by F. Brackx. 1991 Mathematics Subject Classification : 30D45, 46E15. Key words and phrases : bq , qK -sequences, meromorphic functions , Besov classes. Bull. Belg. Math. Soc. Simon Stevin 16 (2009), 1–14 A. El-Sayed Ahmed – M. A. Bakhit 2 center a ∈ ∆ and radius r. For 0 < q < ∞ and 0 < s < ∞, the classes M# ( p, q, s ) are defined in [15] as follows:   ZZ p 2
q 2
s # # M ( p, q, s ) = f ∈ M(∆) : sup ( f (z)) 1 − |z| 1 − | ϕ a (z)| dA(z) < ∞ , ∆ a∈∆ where f # (z) = | f ′ (z)| 1+| f (z)|2 is the spherical derivative of f . The classes M# (q, q are called the Besov-type classes, they are denoted by Bq# , where Bq#  = f ∈ M(∆) : sup a∈∆ ZZ ∆ f # (z)
q 1 − |z| 2
q −2 (1) − 2, 0) dA(z) < ∞ . But in this paper the meromorphic Besov-type classes always refer to the classes M# (q, q − 2, s). Let 0 < q < ∞ and 0 < s < ∞. Then the Besov-type classes are defined by: # Bq,s =  f ∈ M(∆) : sup a∈∆ ZZ ∆ f # (z)
q 1 − |z|2
q −2
s 1 − | ϕ a (z)| 2 dA(z) < ∞ , 2
q −2 2
s (2) where the weight function is (1 − |z| 1 − | ϕ a (z)| and z ∈ ∆. For more information about holomorphic and meromorphic Besov classes, we refer to [5, 6, 10, 11, 12, 14, 17, 18, 23] and others. Recently Wulan [20] gave the following definition: Definition 1.1. Let K : [0, ∞) → [0, ∞) be a nondecreasing function. A function f meromorphic in ∆ is said to belong to the class Q#K if sup a∈∆ ZZ ∆
2
f # (z) K g(z, a) dA(z) < ∞, where, the function g(z, a) = ln 1a−−āz z is defined as the composition of the Möbius transformation ϕ a and the fundamental solution of the two-dimensional real Laplacian. Q#K space has been studied during the last few years (see e.g [8, 9] and others). The meromorphic counterpart of the Bloch space is the class of normal functions N (see [1, 2, 15, 16, 21]), which is defined as follows: Definition 1.2. Let f be a meromorphic function in ∆. If k f kN = sup(1 − |z|2 ) f # (z) < ∞, (3) z∈∆ then f belongs to the class N of normal functions. Definition 1.3. ([4]) Let f be a meromorphic function in ∆. A sequence of points { an } (| an | → 1) in ∆ is called a qN −sequence if lim f # (an )(1 − | an |2 ) = +∞. n→∞ Now, we will introduce the following definitions: (4) Sequences of some meromorphic function spaces 3 Definition 1.4. Let f be a meromorphic function in ∆, 2 < q < ∞ and 0 < s < ∞. A sequence of points { an }(| an | → 1) in ∆ is called a bq−sequence if lim n→∞ ZZ ∆
q
s f # (z) (1 − |z|2 )q−2 1 − | ϕ an (z)| 2 dA(z) = +∞. (5) Definition 1.5. Let f be a meromorphic function in ∆. For a function K, K : [0, ∞) −→ [0, ∞). A sequence of points {an }(| an | → 1) in ∆ is called a qK −sequence if lim n→∞ 2 ZZ ∆
2 f # (z) K ( g(z, an ))dA(z) = +∞. (6) bq and qN −sequences In this section, we study some new sequences of some meromorphic function spaces such as bq and qN −sequences. Our study is motivated by the theories of normal and meromorphic Besov functions. We prove various results about these sequences. For example, if { an } is a q N sequence for the meromorphic function f and {bn } is a sequence with σ(an , bn ) → 0 as n → ∞, where σ denotes the pseudohyperbolic distance, then {bn } is a bq sequence for f for every q > 2. We will need the following definition in the sequel: Definition 2.1. [19] Let f be a meromorphic function in C. If the family { f (z + an )} is normal for any sequence { an } of complex numbers, then f is a Yosida function y(z). Theorem 2.1. Let f be a meromorphic function in ∆. If { an } is a qN −sequence, then any sequence of points {bn } in ∆ for which σ(an , bn ) → 0 is a bq−sequence for all q, 2 < q < ∞. Proof. By([19], theorem 4.4.1) with β = 0 and α = 1, there exist sequences {bn } ⊂ ∆ and { pn } ⊂ R + , with σ(an , bn ) −→ 0 and pn −→ 0, (1 − | b n | 2 ) (7) where the sequence of functions { f n (t)} = { f (bn + pn t)} converges uniformly on each compact subset of C to a nonconstant Yosida function y(t). Then sup b ∈∆ ≥ ZnZ ≥ ZZ ZZ ∆
s ( f # (z))q (1 − |z|2 )q−2 1 − | ϕbn (z)| 2 dA(z) ∆ (bn , 1e ) ∆ (0,r )
s ( f # (z))q (1 − |z|2 )q−2 1 − | ϕbn (z)| 2 dA(z)
s 2− q (y#n (t))q (1 − |bn + pn t|2 )q−2 1 − | ϕbn (bn + pn t)| 2 pn dA(t) 2 s bn − ( bn + p n t ) = dA(t) × 1− 1 − b̄n (bn + pn t) ∆ (0,r )   ZZ 2 s 2  q −2 1 # q 1 − | bn + p n t | = (yn (t)) × 1 − 1−|b |2 dA(t) . n pn ∆ (0,r ) − b̄ n pn t ZZ |y#n (t)|q  1 − | bn + p n t | 2 pn q −2  A. El-Sayed Ahmed – M. A. Bakhit 4 By the uniformly convergence, we have ZZ ∆ (0,r ) ( f n# (t))q dA(t) −→ ZZ ∆ (0,r ) (y# (t))q dA(t), and this last integral is positive, because y(t) is a nonconstant meromorphic function. Moreover, using (7) as n → ∞, we obtain that 1− 2 1 1−| bn |2 pn t − b̄n −→ 1. Then, we conclude that   ZZ 2  q −2 # q 1 − | an + pn t| × 1− (yn (t)) pn ∆ (0,r ) 1 1−| an |2 pn t − ān 2 s dA(t) −→ ∞, and it follows for all q, where 2 < q < ∞ that ZZ ∆
s ( f # (z))q (1 − |z|2 )q−2 1 − | ϕbn (z)| 2 dA(z) −→ ∞, then {bn } ∈ ∆ is a bq −sequence for all q, 2 < q < ∞. Thus the proof of Theorem 2.1 is established. Theorem 2.2. There exist a non-normal function f and { an } in ∆ which is a bq −sequence for all q, 2 < q < ∞, but { an } is not a qN −sequence. Proof. By ([4] theorem 2), we can consider a function f (z) = exp ( 1−i z ) be not √ 2 normal, i = −1 . Choose a sequence {bn } = { 1+n n2 } and by a computation, we obtain that lim (1 − |bn |2 ) f # (bn ) = +∞. n→∞ By Theorem 2.1 for any sequence of points { an } in ∆ for which σ(an , bn ) → 0, lim n→∞ ZZ ∆
s ( f # (z))q (1 − |z|2 )q−2 1 − | ϕ an (z)| 2 dA(z) = +∞, 2 for all q, 2 < q < ∞. Now we choose { an } = { 1+n n2 − σ(an , bn ) → 0. But lim (1 − | an |2 ) f # (an ) = 0. i } n + n3 and notice that n→∞ Thus { an } is just one we need. Theorem 2.3. Let f be a meromorphic function in ∆ and let 2 < q′ < q < ∞ and 0 < s′ < s < ∞. If, for a sequence of points { an } in ∆, lim n→∞ then lim n→∞ ZZ ZZ ∆
s ( f # (z))q (1 − |z|2 )q−2 1 − | ϕ an (z)| 2 dA(z) = +∞, ′ ∆ ′ ( f # (z))q (1 − |z|2 )q −2 1 − | ϕ an (z)| 2
s′ dA(z) = +∞. (8) (9) Sequences of some meromorphic function spaces 5 Proof. If assumption (8) holds for 2 < q′ < q < ∞ and 0 < s′ < s < ∞, then by Hölder’s inequality, we have that ZZ ′ ∆ ≤ Z Z × Z Z = Z Z × Z Z ∆ ( f # (z))q (1 − |z|2 )q−2 1 − | ϕ an (z)| ′ 2 (s − ∆ ∆ ∆ (1 − | ϕ an (z)| ) ∆ (1 − | w | 2 ) ( sq ′ q q )( q − q ′ ) (1 − | w | 2 ) s ′ q − sq ′ − 2) q−q′ Thus, s′ q − sq ′ q−q′ s ′ q − sq ′ ( q − q ′ − 2) dA(w)
s′ dA(z) 2
s dA(z)  (1− qq′ ) 2
s dA(z) ZZ ∆  (1− qq′ )  qq′ . − 2) = (κ − 2) > −1, for κ = dA(w) =  qq′ (1 − |z|2 )−2 dA(z) ( f # (z))q (1 − |z|2 )q−2 1 − | ϕ an (z)| Since it is easy to check ( obtain that ZZ ′ ( f # (z))q (1 − |z|2 )q −2 1 − | ϕ an (z)| 2 s′ q − sq ′ q−q′ > 1 , then we (1 − |w|2 )(κ−2) dA(w) < C < ∞, for C > 0. M# (q, q − 2, s) ⊂ M# (q′ , q′ − 2, s′ ), Hence, we obtain that ZZ ≥ ′ Z Z∆ ∆ ′ ( f # (z))q (1 − |z|2 )q −2 1 − | ϕ an (z)| 2
s′ dA(z)
s ( f # (z))q (1 − |z|2 )q−2 1 − | ϕ an (z)| 2 dA(z) = +∞. Then assumption (9) holds. Hence the proof of Theorem 2.3 is completed. Remark 2.1. By assumption (8), we know that f ∈ / M# (q, q − 2, s). Since the function classes M# (q, q − 2, s) have a nesting property, f ∈ / M# (q′ , q′ − 2, s′ ), for q′ < q and 0 < s′ < s < ∞. However, Theorem 2.3 gives more information about this situation showing that the same sequence { an }, which breaks the M# (q, q − 2, s)−condition, also breaks M# (q′ , q′ − 2, s′ )−condition. Remark 2.2. In fact, from the proof of Theorem 2.3, we can see that if for a fixed r0 , 0 < r0 < 1 and R > 0, lim n→∞ ZZ ∆ ( an ,r0 )
s ( f # (z))q (1 − |z|2 )q−2 1 − | ϕ an (z)| 2 dA(z) = +∞,
then there exists a sequence of points {bn } in URn = {z : (1 − | ϕ a (z)| 2 > R}, such that lim (1 − |bn |2 ) f # (bn ) = +∞. n→∞ A. El-Sayed Ahmed – M. A. Bakhit 6 Theorem 2.4. Let f be a meromorphic function in ∆. If, for a sequence of points { an } in ∆, lim (1 − | an |2 ) f # (an ) = +∞, (10) n→∞ then for the same sequence { an } lim n→∞ ZZ ∆ ( an ,r )
s ( f # (z))q (1 − |z|2 )q−2 1 − | ϕ an (z)| 2 dA(z) = +∞, holds for all q, s, 2 < q < ∞ , 0 < s < ∞ and all r, 0 < r < 1. Proof. Suppose that (10) holds. If there exists an r0 , 0 < r0 < 1 and p, 1 < p < ∞, such that lim sup n→∞ ZZ ∆ ( an ,r0 )
s ( f # (z))q (1 − |z|2 )q−2 1 − | ϕ an (z)| 2 dA(z) = M < +∞, then there exists a subsequence { ank } of { an }, such that ZZ ∆ ( ank ,r0 )
s ( f # (z))q (1 − |z|2 )q−2 1 − | ϕ ank (z)| 2 dA(z) ≤ M + 1, for k sufficiently large. Now, choose an r1 , 0 < r1 < r0 , ∆(ank , r1 ) = {z ∈ ∆ :| ϕ ank (z) |< r1 }, satisfying π M+1 < . 2 s + q − 2 2 (1 − r 1 ) It follows that ZZ ∆ ( ank ,r1 ) ( f # (z))q dA(z) ≤ M+1 π < , 2 s + q − 2 2 (1 − r 1 ) for (1 − | ϕ ank (z)|2 ) ≥ (1 − r12 ). By Dufresngy’s theorem (see [16] pp.83 ), we have (1 − | ank |2 ) f # (ank ) ≤ r1 , which 1 contradicts our assumption. Hence the proof of Theorem 2.4 is completed. Theorem 2.5. Let f be a meromorphic function in ∆. Suppose for 0 < p < ∞, there exists a sequence of points { an } ⊂ ∆, such that lim n→∞ ZZ ∆ ( f # (z))q (1 − |z|2 )q−2 (1 − | ϕ an (z)|2 )s dA(z) = +∞. Then, for any sequence of points {bn } in ∆ for which σ(an , bn ) → 0, lim n→∞ ZZ ∆
s ( f # (z))q (1 − |z|2 )q−2 1 − | ϕbn (z)| 2 dA(z) = +∞. Proof. Choose positive constants M1 and M2 such that M2 < M1 . Let n n UM = {z : (1 − | ϕ an (z)|2 ) > M1 } and U M = {z : (1 − | ϕ an (z)|2 ) > M2 }. 2 1 Sequences of some meromorphic function spaces 7 n , z ∈ ∆\U n and C (1 − | ϕ (z)|2 ) ≤ (1 − | ϕ (w)|2 for some Then if w ∈ U M an an M2 1 constant C > 0. This means for all n that, ZZ
s ( f # (z))q (1 − |z|2 )q−2 1 − | ϕbn (z)| 2 dA(z) n ∆ \U M ≥ Cs Z Z2 n ∆ \U M 2
s ( f # (z))q (1 − |z|2 )q−2 1 − | ϕ an (z)| 2 dA(z), (11) for any sequence of points {bn } in ∆ for which σ(an , bn ) → 0. If ZZ
s lim sup ( f # (z))q (1 − |z|2 )q−2 1 − | ϕ an (z)| 2 dA(z) = +∞, n→∞ n ∆ \U M 2 Then, by (11) lim sup n→∞ ZZ n ∆ \U M 2
s ( f # (z))q (1 − |z|2 )q−2 1 − | ϕbn (z)| 2 dA(z) = +∞. Also, if lim sup n→∞ ZZ n UM 2
s ( f # (z))q (1 − |z|2 )q−2 1 − | ϕ an (z)| 2 dA(z) = +∞, then, we have two different cases: n for which σ ( a , c ) → 0, Either (i) there exists a sequence of points {cn } in U M n n 2 such that lim (1 − |cn |2 ) f # (cn ) = +∞, n→∞ or (ii) there exists r0 , 0 < r0 < e− M2 and K > 0, such that (1 − |z|2 ) f # (z) ≤ K, for all z ∈ ∆(an , r0 ). If (i) is true, then, by Theorem 2.1, for above {bn }, for which σ(an , bn ) → 0, ZZ
s lim sup ( f # (z))q (1 − |z|2 )q−2 1 − | ϕbn (z)| 2 dA(z) = +∞ , n→∞ ∆ since σ(bn , cn ) → 0. On the other hand, if (ii) holds, then using the same conclusions for weight functions we see that necessarily for any sequence of points {bn } for which σ(an , bn ) → 0, ZZ
s lim sup ( f # (z))q (1 − |z|2 )q−2 1 − | ϕbn (z)| 2 dA(z) = +∞. n→∞ ∆ This completes the proof. Now, we consider the following question. Question 2.1 Let 1 < q < ∞ for any sequence of points { an } and suppose that ZZ
s lim ( f # (z))q (1 − |z|2 )q−2 1 − | ϕ an (z)| 2 dA(z) = +∞. n→∞ Is it true for q′ , where q < q′ , lim sup n→∞ ∆ ZZ ∆
s ′ ′ ( f # (z))q (1 − |z|2 )q −2 1 − | ϕ an (z)| 2 dA(z) = +∞? We answer the question by Theorem 2.6. A. El-Sayed Ahmed – M. A. Bakhit 8 Definition 2.2. Let 2 < q < ∞. For any sequence of points { an } in ∆ is a mq −sequence if ZZ
q lim sup ( f # (z))q (1 − |z|2 )q−2 1 − | ϕ an (z)| 2 dA(z) = +∞. n→∞ ∆ Our answer to Question 2.1 is naturally as follows: Theorem 2.6. Let 2 < q < ∞ and suppose that lim n→∞ ZZ ∆ ( f # (z))q (1 − |z|2 )q−2 (1 − | ϕ an (z)|2 )s dA(z) = +∞. If the sequence of points { an } in ∆ is not a mq −sequence, then for any q′ and q < q′ with q′ + s > 1, then we have lim n→∞ ZZ ∆
s ′ ′ ( f # (z))q (1 − |z|2 )q −2 1 − | ϕ an (z)| 2 dA(z) = +∞. Proof. Since, (i ) M# (q, q − 2, s) ⊂ N for all q, 2 < q < ∞ and 0 < s < 1 (see [15] theorem 3.3.3). (ii ) [ 2< q < q ′ M# (q, q − 2, s) ( M# (q′ , q′ − 2, s) for all q, where q′ , 2 < q < ∞ and 0 < s < 1 with q′ + s > 1, the proof of this result can be found in [15]. So, it is easy to see that lim n→∞ 3 ZZ ∆
s ′ ′ ( f # (z))q (1 − |z|2 )q −2 1 − | ϕ an (z)| 2 dA(z) = +∞. qK and qN -sequences Now, we study bq and qN -sequences. We prove many results about these sequences. Our results are obtained by the help of normal and Q#K functions. For example, if { an } is a q N sequence for the meromorphic function f and {bn } is a sequence with σ(an , bn ) → 0 as n → ∞, where σ denotes the pseudohyperbolic distance, then {bn } is a qK sequence for f . Now, we give the following theorem: Theorem 3.1. Let f be a meromorphic function in ∆. If { an } is a qN −sequence, then any sequence of points {bn } in ∆ for which σ(an , bn ) → 0 is a qK −sequence for all K, K (t) → ∞ as t → ∞. Proof. By ([7], theorem 7.2), there exist a sequences {cn } ⊂ ∆ and { pn } ⊂ R + , with pn −→ 0, (12) σ(an , cn ) −→ 0 and (1 − | c n | 2 ) where the sequence of functions { f n (t)} = { f (cn + pn t)} converges uniformly on each compact subset of C to a nonconstant meromorphic function y(t). For a fixed R > 0 set ∆n = {z : z = cn + pn t, |t| < R}. Now, for any sequence Sequences of some meromorphic function spaces 9 of points {bn } ⊂ ∆, for which σ(an , cn ) −→ 0, we have σ(bn , cn ) −→ 0 since σ(an , cn ) −→ 0. Thus, for n sufficiently large, we obtain that ∆n = {z : z = cn + pn t, |t| < R} ⊂ Ωn = {z : | ϕbn (z)| < 1 }. e Therefore, we get by change of variables ZZ ≥ ZZ = ZZ Ωn ∆n ( f # (z))2 K ( g(z, bn )dA(z) ( f # (z))2 K ( g(z, bn )dA(z) | t|< R ( f # (z))2 K ( g(cn + pn t, bn )dA(z). By the uniformly convergence, we have ZZ | t|< R ( f n# (t))2 dA(t) −→ ZZ | t|< R (y# (t))2 dA(t), and the last integral is finite and non-zero, because y(t) is a non-constant meromorphic function. However, g(cn + pn t, bn ) → +∞ as n → +∞ uniformly, for |t| < R, we obtain that ZZ | t|< R (y# (z))2 K ( g(cn + pn t, bn )dA(z) −→ ∞. In fact, g(cn + pn t, bn ) = log 1 − bn ( c n + p n t ) c n + p n t − bn Moreover, using (12) as n → ∞, we obtain that c n + p n t − bn 1 − bn ( c n + p n t ) ≤ ≤ | c n − bn | + p n | t | | 1 − bn c n | − p n | bn t | cn − bn 1− b n c n 1− + |1−pnb|tc| n n| pn | t| | 1− b n c n | −→ 0. For all K, K (t) → ∞ as t → ∞, it follows that ZZ ∆ ( f # (z))2 K ( g(z, bn )dA(z) −→ ∞, then {bn } ∈ ∆ is a qK −sequence for all K. Thus the proof of Theorem 3.1 is therefore established. Theorem 3.2. There exist a non-normal function f and { an } in ∆ which is a qK −sequence for all K, K : [0, ∞) → [0, ∞), but { an } is not a qN −sequence. Proof. The proof of this theorem is much akin to the proof of Theorem 2.2. So, it will be omitted. A. El-Sayed Ahmed – M. A. Bakhit 10 4 Non-normal functions and æN −sequences. In this section we define the concept of ρN −sequences of meromorphic functions which allows one to describe non-normal functions. We give the necessary and sufficient condition for the sequence of points {zn }, where limn→∞ |zn | = 1 to be a ρN −sequence in terms of the growth of f . Makhmutov defined the concept of ρB −sequences of holomorphic functions f (z) in the unit disk ∆ (see [13], pp. 9 definition 5.2.) as follows: Definition 4.1. A sequence of points {zn }, limn→∞ |zn | = 1, is a ρB −sequence of holomorphic functions f (z) ∈ ∆, if there are two sequences of numbers {ε n }, where limn→∞ |ε n | = 0 and { Mn }, limn→∞ Mn = ∞, for which the diameter of f (∆(zn , ε n )) exceeds { Mn } for each n. Now, we define ρN −sequences of meromorphic functions. Definition 4.2. A sequence of points {zn } with limn→∞ |zn | = 1, is a ρN −sequence of meromorphic functions f , if there are two sequences of numbers {ε n }, where limn→∞ |ε n | = 0 and { Mn }, limn→∞ Mn = ∞, for which the diameter of f (∆(an , ε n )) exceeds { Mn } for each n. Now, we let A f (a, r ) = ZZ ∆ ( a,r ) ( f # (z))2 dxdy be the area of the Riemann image of ∆(a, r ) by f and L(a, r ) = ZZ ∆ ( a,r ) f # (z) |dz| be the length of the Riemann image of the pseudohyperbolic circle Γ(a, r ) by f . Let F(a, r ) be the Riemann image of ∆(a, r ) by f and F (a, r ) be the projection of F(a, r ) to C. Let A f (a, r ) be the Euclidean area of F (a, r ) and L(a, r ) be the length of the outer boundary of F (a, r ). It is clear that A f (a, r ) ≤ A f (a, r ) and L f (a, r ) ≤ L f (a, r ) for each a ∈ ∆ and each 0 < r < 1. Yamashita proved in [22] that, for any holomorphic function f (z) or a meromorphic function f in ∆, any a ∈ ∆ and 0 < r < 1, (1 − | a | 2 ) f # ( a ) ≤  A f (a, r ) πr2  12 , L f (a, r ) . 2πr Now, we give the following important proposition. (1 − | a | 2 ) f # ( a ) ≤ Proposition 4.1. If f is a meromorphic function in ∆ and {zn }, limn→∞ |zn | = 1, is such that lim (1 − |zn |2 ) f # (zn ) = +∞, n→∞ then {zn } is a ρN −sequence of the meromorphic function f . Sequences of some meromorphic function spaces 11 Proof. Suppose that f is a meromorphic function in ∆ and {zn }, limn→∞ |zn | = 1, lim (1 − |zn |2 ) f # (zn ) = +∞, n→∞ let (1 − | z n | 2 ) = ε n and M n = f # ( z n ), then there are two sequences of numbers {ε n }, Mn where lim |ε n | = 0 and lim Mn = 0. n→∞ n→∞ By Definition 4.2, we have a sequence of points {zn } which is a ρN −sequence. If the sequence of points {zn } is a ρN −sequence of the meromorphic function f , then there are two sequences limn→∞ (1 − |zn |2 ) = 0 as limn→∞ |zn | = 1 and limn→∞ f # (zn ) = +∞. Our proposition is therefore proved. Theorem 4.1. A meromorphic function f is not a normal function if and only if it has a ρN −sequence of points. Proof. Necessity. If f ∈ / N , then there exists a sequence {zn } which satisfies the condition lim (1 − |zn |2 ) f # (zn ) = +∞. n→∞ By Proposition 4.1, the sequence {zn } is a ρN −sequence of the meromorphic function f . Sufficiency. Let { an } be a ρN −sequence of the meromorphic function f . If f ∈ N by ([13] theorem 3.4) we have L f (a, r ) and A f (a, r ) are bounded for any 0 < r < 1, i.e. the diameters of f (∆(an , r )) don’t tend to infinity. This contradicts our assumption that { an } is a ρN −sequence of f . Theorem 4.2. Let { an } be a ρN −sequence of the meromorphic function f and {bn } be such that lim σ(an , bn ) = 0, (13) n→∞ then {bn } is a ρN −sequence of f too. Proof. Let { an } be a ρN −sequence of the meromorphic function f and {bn } be not a ρN −sequence of f . Then by Definition 4.2 for each δ > 0, we have lim A f (bn , δ) < ∞, n→∞ and lim L f (bn , δ) < ∞. n→∞ Suppose ε = 2δ . As limn→∞ σ(an , bn ) = 0, then beginning with some N for any n > N, we obtain ∆(an , ε) ⊂ ∆(bn , δ) and hence , Thus, f (∆(an , ε)) ⊂ f (∆(bn , δ)). dim f (∆(an , ε)) → ∞ as n → ∞, A. El-Sayed Ahmed – M. A. Bakhit 12 which implies that, dim f (∆(bn , δ)) → ∞. This is a contradiction from our hypothesis. Remark 4.1. We need to remind the reader that the pseudohyperbolic circle Γ(zn , ρn ) with center zn and radius ρn is the same as Euclidean circle {z : |z − ẑn | = rn with rn = 1−| zn |2 1−| zn |2 ρ2n 1−| ρn |2 2 2. n | ρn and ẑn = zn 1−|z In particular, ρn → 0 if and only if rn 1−| zn |2 → 0. Now we prove the next theorem : Theorem 4.3. A sequence {zn }, (|zn | → 1), is a ρN −sequence of the meromorphic function f if and only if there is a sequence of positive numbers {ε n }, (ε n → 0) such that lim sup (1 − |z|2 ) f # (z) = +∞. (14) n→∞ z∈∆ (zn ,ε n ) Proof. Necessity. Let {zn } be a ρN −sequence of the meromorphic function f . Then by ([3], Lemma 2), there are sequences { an } and {bn } such that 1 lim σ(an , zn ) = 0, lim σ(bn , zn ) = 0 and lim | f # (an ) − f # (bn )| ≥ . n→∞ n→∞ n→∞ 2 Suppose δn = max{|zn − an |, |zn − bn |} and Ln is a segment connecting the points an and bn . Since an and bn lie in a disk with hyperbolic radius tending to zero then by Remark 4.1, 1−|δnz |2 must also tend to zero. For some wn ∈ Ln , we have n that # | a n − bn | f ( w n ) ≥ Z # Ln f (z)|dz| ≥ Z Ln f # (z)dz = f # (an ) − f # (bn ) ≥ 1 . 2 On the other hand for sufficiently large n, we have that (1 − | w n | 2 ) f # ( w n ) ≥ (1 − | w n | 2 ) 1 − (|zn | + δn )2 1 ≥ = 2 | a n − bn | 4δn 1 − |zn |2 |zn | δn − − . 4δn 2 4 The last expression tends to ∞ and condition (14) is proved . Sufficiency. Let {zn } be such sequence of points that lim (1 − |zn |2 ) f # (zn ) = +∞, n→∞ {ε n } be a sequence of positive numbers, limn→∞ ε n = 0 and zn ∈ ∆(zn , ε n ) for each n. 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Sohag University Faculty of Science, Department of Mathematics, Sohag, Egypt e-mail: [email protected]