Normality of Meromorphic Functions and Uniformly Discrete Exceptional Sets
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Normality of Meromorphic Functions and Uniformly Discrete Exceptional Sets Jianming Chang
Received: 11 May 2012 / Revised: 10 September 2012 / Accepted: 8 December 2012 / Published online: 23 January 2013 © The Author(s) 2013. This article is published with open access at Springerlink.com
Abstract Let k ∈ N and h(≡ 0) be a function holomorphic on D. Let F be a family of meromorphic functions in D, all of whose zeros have multiplicity at least k + 3. Suppose that the sets {E f } f ∈F are locally uniformly discrete in D, where E f = {z ∈ D : f (z) = 0} ∪ {z ∈ D : f (k) (z) = h(z)}. Suppose additionally that at the common zeros of f ∈ F and h, the multiplicities m f for f and m h for h satisfy m f ≥ m h + k + 1 for k > 1 and m f ≥ 2m h + 3 for k = 1. Then, F is normal in D. The number k + 3 can be replaced by k + 2 if the set E f is independent of f , or in other words, for each pair of functions f and g in F, f and g share the value 0, and f (k) and g (k) share the function h. Examples are also given to show that the conditions are necessary and sharp. Keywords Meromorphic function · Normal family · Shared values · Locally uniformly discrete sets Mathematics Subject Classification (2000)
Primary 30D45
1 Introduction and Main Results A family F of meromorphic functions defined in a plane domain D ⊂ C is said to be normal in D, if each sequence { f n } ⊂ F contains a subsequence which converges spherically locally uniformly in D to a meromorphic function or ∞. See [6,11,15]. Communicated by Lawrence Zalcman. Research supported by NNSF of China, Grant No. 11171045. J. Chang (B) Department of Mathematics, Changshu Institute of Technology, Changshu, Jiangsu 215500, People’s Republic of China e-mail: [email protected]
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The Gu’s normality criterion [5] which was conjectured by Hayman [6] says that a family F of functions meromorphic on D is normal if f = 0 and f (k) = 1 for each f ∈ F. The following generalization of Gu’s theorem was proved by Yang [14]. Theorem 1 Let F be a family of meromorphic functions on D, k ∈ N and h(≡ 0) be a holomorphic function on D. If for every f ∈ F, f = 0 and f (k) = h on D, then F is normal on D. In recent years, following Schwick [12], many normality criteria concerning shared values or functions have been proved. We say that two functions f and g share a value or a function φ if the two equations f (z) = φ(z) and g(z) = φ(z) have the same solutions (ignoring multiplicity). Here, we want to generalize the following result of Fang and Zalcman [4] by replacing the constant 1 by a function. Theorem 2 Let k be a positive integer and let F a family of meromorphic functions on D, all of whose zeros have multiplicity at least k + 2, such that for each pair of functions f and g in F, f and g share the value 0, and f (k) and g (k) share the value 1. Then, the family F is normal. In general, the constant 1 cannot be replaced by a function. For example, the family { f n }, where f n (z) = nz k+2 , is not normal at 0. However, each pair of functions f n and f m share the va
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