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Normal Functions and Normal Families BULLETIN of the Bull. Malays. Math. Sci. Soc. (2) 27 (2004), 149−160 MALAYSIAN MATHEMATICAL SCIENCES SOCIETY Normal Functions and Normal Families YAN XU Department of Mathematics, Nanjing Normal University, Nanjing 210097 e-mail: [email protected] Abstract. In this paper, we prove the following theorem: Let F be a family of holomorphic functions in the unit disc D and let a be a nonzero complex number. If, for any f ∈ F , f (z) = a ⇒ f ′(z) = a , f ′(z) = a ⇒ f ′′(z) = a , then F is uniformly normal in D, that is, 2 # there exists a positive constant M such that (1 − z ) f (z) ≤ M for each f ∈ F and z ∈ D , where M is independently of f. This result improves related results due to [2], [8], and [3]. 2000 Mathematics Subject Classification: 30D45 1. Introduction Let f and g be two meromorphic functions, and let a be a complex number. If g(z) = a whenever f (z) = a , we denote it by f = a ⇒ g = a , and f = a ⇔ g = a means f (z) = a if and if only if g(z) = a . Let D denote the unit disk in the complex plane C. A function f meromorphic in D is called a normal function, in the sense of [6], if there exist a constant M ( f ) such that (1 − z 2 ) f # (z) ≤ M ( f ) for each z ∈ D , where | f ′(z) | f # (z) = 1+ | f (z) | 2 denotes the spherical derivative. Let F be a family of meromorphic functions defined in D. F is said to be normal in D (see [9]), in the sense of Montel, if for any sequence fn ∈ F there exists a subsequence f , such that f converges spherically, locally and uniformly in D, to a n j n j meromorphic function or ∞ . 150 Y. Xu Suppose that F is a family of functions meromorphic in D such that each function of F is a normal function, then, for each function f ∈ F , there exists a constant M ( f ) such that (1 − z 2 ) f # (z) ≤ M ( f ) for each z ∈ D . In general, M ( f ) is a constant dependent on f, and we can not conclude that {M ( f ) , f ∈ F} is bounded. If {M ( f ) , f ∈ F} is bounded, we give the definition as follows Definition. Let F be a family of meromorphic functions in the unit disc D. If there exists a positive constant M such that sup {(1 − z 2 ) f # (z) : z ∈ D , f ∈ F } < M , we call the family F a uniformly normal family in D. Remark 1. The idea of this definition is suggested by Pang (see [7]). Remark 2. A well-known result due to Marty (see [4], [9] and [11]) says that a family F of functions meromorphic in D is a normal family if and only if for each # compact subset K of D there exists a constant M K such that f (z) ≤ M K for each f ∈ F and for each z ∈ K . Clearly, by Marty’s criterion if F is a uniformly normal family in D , then F must be normal in D . However, it is obvious that the converse is not always true. It is natural to ask: When is a normal family F in D also uniformly normal in D? (The question is first introduced by Bergweiler and Pang (see [7]).) Schwick [10] discovered a connection between normality criteria and sharing values. He proved Theorem A. Let F be a family of meromorphic functions in the unit disc D and let a1 , a2 and a3 be distinct complex numbers. If, for any f ∈ F , f (z) = ai ⇔ f ′(z) = ai (i = 1, 2, 3) , then F is normal in D. Pang [7] proved that the family F in Theorem A is also uniformly normal, as follows Theorem B. Let F be a family of meromorphic functions in the unit disc D and let a1 , a2 and a3 be distinct complex numbers. If, for any f ∈ F , Normal Functions and Normal Families 151 f (z) = ai ⇔ f ′(z) = ai (i = 1, 2, 3) , then F is uniformly normal in D, that is, there exists a positive constant M such that (1 − z 2 ) f # (z) ≤ M for each f ∈ F and z ∈ D , where M is independent of f. Remark 3. In fact, from the proof in [7], we see that Theorem B still remains true if f (z) = ai ⇒ f ′(z) = ai (i = 1, 2, 3) for any f ∈ F . Chen and Hua [2], Pang and Zalcman [8] proved the following normality criterion. Theorem C. Let F be a family of holomorphic functions in the unit disc D and let a be a nonzero complex number. If, for any f ∈ F , f (z) = a ⇔ f ′ = a , f ′ = a ⇒ f ′′(z) = a , then F is normal in D. In [3], Fang improved Theorem C as follows Theorem D. Let F be a family of holomorphic functions in the unit disc D and let a be a nonzero complex number. If, for any f ∈ F, f (z) = a ⇒ f ' (z) = a, f ′(z) = a ⇒ f ′′(z) = a , then F is normal in D. In this paper, by using a method different from that used in [3], we obtain the following stronger result. Theorem 1. Let F be a family of holomorphic functions in the unit disc D and let a be a nonzero complex number. If, for any f ∈ F , f (z) = a ⇒ f ′(z) = a , f ′(z) = a ⇒ f ′′(z) = a , then F is uniformly normal in D, that is, there exists a positive constant M such that (1 − z 2 ) f # (z) ≤ M for each f ∈ F and z ∈ D , where M is independent of f. Remark 4. The following example (see [2] and [3]) shows that a ≠ 0 cannot be omitted in Theorem 1. nz Let F = { fn(z) = e : n = 1, 2, 3"}, D = { z : z < 1}. Then, for every fn ∈ F, it is easy to see that fn(z) = 0 ⇒ fn′(z) = 0 ⇒ fn′′(z) = 0. However, # fn (0) = n / 2 → ∞ as n → ∞ , thus F is not uniformly normal in D. We shall use the standard notations in Nevanlinna theory (see [4], [11]). 152 Y. Xu 2. Lemmas For convenience, we define ⎛ f ′ ⎞ ⎛ f ′′ ⎞ ⎛ f ′ ⎞ LD(r, f ) : = c1m ⎜r, ⎟ + c2m ⎜r, ⎟ + c3m ⎜r, ⎟ ⎝ f ⎠ ⎝ f ′ ⎠ ⎝ f − a ⎠ ⎛ f ′′ ⎞ ⎛ f ′′ ⎞ + c4m ⎜r, ⎟ + c5m ⎜r, ⎟ , (a ∈ C) ⎝ f − a ⎠ ⎝ f ′ − a ⎠ where c1, c2 , c3 , c4 , c5 are constants, which may have different values at different occurrences. Lemma 1. Let f be a non-constant holomorphic functions on the unit disc D, and a be a nonzero complex number. Let f ′(z) + f ′′(z) 2 f ′′(z) ψ (z) := ψ ()f (z) = − . f (z) − a f ′(z) − a If f = a ⇒ f ′ = a , f ′ = a ⇒ f ′′ = a on D, and f (0) ≠ a , f ′(0) ≠ a , f ′′(0) ≠ 0 , f ′(0) ≠ f ′′(0) and ψ (0) ≠ 0 , then f (0) − a T( r, f ) ≤ LD (r, f ) + O (1) + 3log f ′′(0) − f ′(0) ( f (0) − a) ( f ′(0) − a) 1 + log + 2 log . f ′′(0) ψ (0) Proof. Let f (z0 ) = a . By the assumptions we may assume that, near z0 a f (z) = a + a (z − z ) + ()()z − z 2 + b z − z 3 + O ()(z − z )4 , 0 2 0 0 0 (3) where b = f ( z0 ) / 6 is a constant. Then 2 3 f ′(z) = a + a ()z − z0 + 3b ( z − z0 ) + O (( z − z0 ) ), 2 f ′′(z) = a + 6b ()z − z0 + O ()( z − z0 ) , Normal Functions and Normal Families 153 and thus f ′(z) + f ′′(z) 2 6b = + + O (z − z0 ) , f (z) − a z − z0 a 2 f ′′(z) 2 6b = + + O (z − z0 ) . f ′(z) − a z − z0 a Hence ψ (z0 ) = 0 , and ⎛ 1 ⎞ ⎛ 1 ⎞ 1 N ⎜r, ⎟ ≤ N ⎜r, ⎟ ≤ T ()r,ψ + log ⎝ f − a ⎠ ⎝ ψ ⎠ ψ (0) 1 ≤ N ()r,ψ + LD (r, f )+ log (2.1) ψ (0) ⎛ 1 ⎞ 1 = N0 ⎜r, ⎟ + LD ()r, f + log , ⎝ f ′ − a ⎠ ψ (0) where N0( r,1/ ( f ′ − a)) is the counting function for the zeros of f ′ − a which are not zeros of f − a . Since f = a ⇒ f ′ = a, form (2.1) we get ⎛ 1 ⎞ ⎛ 1 ⎞ 1 2N ⎜r, ⎟ ≤ N ⎜r, ⎟ + LD ()r, f + log . (2.2) ⎝ f − a ⎠ ⎝ f ′ − a ⎠ ψ (0) On the other hand, by the assumptions we have ⎛ ⎞ ⎜ ⎟ ⎛ 1 ⎞ ⎜ 1 ⎟ ⎛ f ′′ ⎞ f ′(0) N ⎜ r, ⎟ ≤ N r, ≤ T ⎜r, ⎟ + log + O (1) ⎝ f ′ − a ⎠ ⎜ f ′′ ⎟ ⎝ f ′ ⎠ f ′′(0) − f ′(0) ⎜ − 1⎟ ⎝ f ′ ⎠ ⎛ f ′′ ⎞ f ′(0) = N ⎜r, ⎟ + log + LD ()r, f + O (1) (2.3) ⎝ f ′ ⎠ f ′′()0 − f ′(0) ⎛ 1 ⎞ f ′(0) = N ⎜r, ⎟ + log + LD ()r, f + O (1). ⎝ f ′ ⎠ f ′′(0) − f ′(0) 154 Y. Xu 1 Next we need the estimate of N (r , f ′ ) . Since ⎛ 1 ⎞ ⎛ 1 ⎞ m ⎜ r, ⎟ ≤ m ⎜r, ⎟ + LD ()r, f ⎝ f − a ⎠ ⎝ f ′ ⎠ ⎛ 1 ⎞ 1 ≤ T ()r, f ′ − N ⎜r, ⎟ + LD ()r, f + log ⎝ f ′ ⎠ f ′(0) ⎛ 1 ⎞ 1 ≤ T ()r, f − N ⎜ r, ⎟ + LD ()r, f + log ⎝ f ′ ⎠ f ′(0) ⎛ 1 ⎞ ⎛ 1 ⎞ f (0) − a ≤ T ⎜ r, ⎟ − N ⎜ r, ⎟ + LD ()r, f + O (1) + log ⎝ f − a ⎠ ⎝ f ′ ⎠ f ′(0) ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎜ = m ⎜ r, ⎟ + N ⎜ r, ⎟ − N ⎜r, ⎟ + LD ( r, f ) + O (1) ⎝ f − a ⎠ ⎝ f − a ⎠ ⎝ f ′ ⎠ f (0) − a + log , f ′(0) we obtain ⎛ 1 ⎞ ⎛ 1 ⎞ f (0) − a N ⎜ r, ⎟ ≤ N ⎜ r, ⎟ + LD (r, f ) + O (1) + log . (2.4) ⎝ f ′ ⎠ ⎝ f − a ⎠ f ′(0) Thus, from (2.2), (2.3) and (2.4) , we get ⎛ 1 ⎞ f (0) − a 1 N ⎜ r, ⎟ ≤ LD (r, f ) + O (1) + log + log , (2.5) ⎝ f − a ⎠ f ′′(0) − f ′(0) ψ (0) ⎛ 1 ⎞ f (0) − a 1 N ⎜ r, ⎟ ≤ LD ( r, f ) + O (1) + 2 log + log .
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