Normal families and shared values

Xuecheng Pang; Lawrence Zalcman

doi:10.1112/s002460939900644x

Normal families and shared values

Xuecheng Pang, Lawrence Zalcman

Research output: Contribution to journal › Article › peer-review

242 Scopus citations

Abstract

For f a meromorphic function on the plane domain D and a ∈ ℂ, let Ē_f(a) = {z ∈ D : f(z) = a}. Let F be a family of meromorphic functions on D, all of whose zeros are of multiplicity at least k. If there exist b ≠ 0 and h > 0 such that for every f ∈ F, Ē_f(0) = Ē_f(k)(b) and 0 < |f^{(k + 1)}(z)| ≤ h whenever = ∈ Ē_f(0), then F is a normal family on D. The case Ē_f(0) = ∅ is a celebrated result of Gu [5].

Original language	English
Pages (from-to)	325-331
Number of pages	7
Journal	Bulletin of the London Mathematical Society
Volume	32
Issue number	3
DOIs	https://doi.org/10.1112/s002460939900644x
State	Published - May 2000
Externally published	Yes

Bibliographical note

Funding Information:
The research of the first author was supported by NNSF of China, Approved No. 19771038, and by the Research Institute for Mathematical Sciences, Bar-Ilan University.

Funding

The research of the first author was supported by NNSF of China, Approved No. 19771038, and by the Research Institute for Mathematical Sciences, Bar-Ilan University.

Funders	Funder number
National Natural Science Foundation of China	19771038
Bar-Ilan University
Research Institute for Mathematical Sciences

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title = "Normal families and shared values",

abstract = "For f a meromorphic function on the plane domain D and a ∈ ℂ, let Ēf(a) = {z ∈ D : f(z) = a}. Let F be a family of meromorphic functions on D, all of whose zeros are of multiplicity at least k. If there exist b ≠ 0 and h > 0 such that for every f ∈ F, Ēf(0) = Ēf(k)(b) and 0 < |f(k + 1)(z)| ≤ h whenever = ∈ Ēf(0), then F is a normal family on D. The case Ēf(0) = ∅ is a celebrated result of Gu [5].",

author = "Xuecheng Pang and Lawrence Zalcman",

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AB - For f a meromorphic function on the plane domain D and a ∈ ℂ, let Ēf(a) = {z ∈ D : f(z) = a}. Let F be a family of meromorphic functions on D, all of whose zeros are of multiplicity at least k. If there exist b ≠ 0 and h > 0 such that for every f ∈ F, Ēf(0) = Ēf(k)(b) and 0 < |f(k + 1)(z)| ≤ h whenever = ∈ Ēf(0), then F is a normal family on D. The case Ēf(0) = ∅ is a celebrated result of Gu [5].

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Normal families and shared values

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