In:
Abstract and Applied Analysis, Hindawi Limited, Vol. 2011 ( 2011), p. 1-8
Abstract:
We study the normality of families of holomorphic functions. We prove the following result. Let α ( z ) , a i ( z ) , i = 1,2 , … , p , be holomorphic functions and F a family of holomorphic functions in a domain D , P ( z , w ) : = ( w - a 1 ( z ) ) ( w - a 2 ( z ) ) ⋯ ( w - a p ( z ) ) , p ≥ 2 . If P w ∘ f ( z ) and P w ∘ g ( z ) share α ( z ) IM for each pair f ( z ) , g ( z ) ∈ F and one of the following conditions holds: (1) P ( z 0 , z ) - α ( z 0 ) has at least two distinct zeros for any z 0 ∈ D ; (2) there exists z 0 ∈ D such that P ( z 0 , z ) - α ( z 0 ) has only one distinct zero and α ( z ) is nonconstant. Assume that β 0 is the zero of P ( z 0 , z ) - α ( z 0 ) and that the multiplicities l and k of zeros of f ( z ) - β 0 and α ( z ) - α ( z 0 ) at z 0 , respectively, satisfy k ≠ l p , for all f ( z ) ∈ F , then F is normal in D . In particular, the result is a kind of generalization of the famous Montel's criterion. At the same time we fill a gap in the proof of Theorem 1.1 in our original paper (Wu et al., 2010).
Type of Medium:
Online Resource
ISSN:
1085-3375
,
1687-0409
Language:
English
Publisher:
Hindawi Limited
Publication Date:
2011
detail.hit.zdb_id:
2064801-7
SSG:
17,1
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