Differential inequalities, normality and quasi-normality

Xiao Jun Liu; Shahar Nevo; Xue Cheng Pang

doi:10.1007/s10114-014-2542-8

Differential inequalities, normality and quasi-normality

Xiao Jun Liu
, Shahar Nevo
, Xue Cheng Pang

Department of Mathematics

Bar-Ilan University

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

Abstract

We prove that if D is a domain in ℂ, α > 1 and C > 0, then the family F of functions f meromorphic in D such that {pipe}f′(z)/1+{pipe}f(z){pipe}^α > C for every z ∈ D is normal in D. For α = 1, the same assumptions imply quasi-normality but not necessarily normality.

Original language	English
Pages (from-to)	277-282
Number of pages	6
Journal	Acta Mathematica Sinica, English Series
Volume	30
Issue number	2
DOIs	https://doi.org/10.1007/s10114-014-2542-8
State	Published - Feb 2014

Keywords

Normal family
differential inequality
quasi-normal family

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1007/s10114-014-2542-8

Cite this

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title = "Differential inequalities, normality and quasi-normality",

abstract = "We prove that if D is a domain in ℂ, α > 1 and C > 0, then the family F of functions f meromorphic in D such that \{pipe\}f′(z)/1+\{pipe\}f(z)\{pipe\}α > C for every z ∈ D is normal in D. For α = 1, the same assumptions imply quasi-normality but not necessarily normality.",

keywords = "Normal family, differential inequality, quasi-normal family",

author = "Liu, \{Xiao Jun\} and Shahar Nevo and Pang, \{Xue Cheng\}",

note = "Funding Information: Received September 10, 2012, revised December 19, 2012, accepted April 16, 2013 The first author is supported by National Natural Science Foundation of China (Grant No. 11071074), the Tianyuan Special Funds of the National Natural Science Foundation of China (Grant No. 11226095), Outstanding Youth Foundation of Shanghai (Grant No. slg10015); the second author is supported by the Israel Science Foundation (Grant No. 395/07); the third author is supported by National Natural Science Foundation of China (Grant No. 11071074)",

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doi = "10.1007/s10114-014-2542-8",

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T1 - Differential inequalities, normality and quasi-normality

AU - Liu, Xiao Jun

AU - Nevo, Shahar

AU - Pang, Xue Cheng

N1 - Funding Information: Received September 10, 2012, revised December 19, 2012, accepted April 16, 2013 The first author is supported by National Natural Science Foundation of China (Grant No. 11071074), the Tianyuan Special Funds of the National Natural Science Foundation of China (Grant No. 11226095), Outstanding Youth Foundation of Shanghai (Grant No. slg10015); the second author is supported by the Israel Science Foundation (Grant No. 395/07); the third author is supported by National Natural Science Foundation of China (Grant No. 11071074)

PY - 2014/2

Y1 - 2014/2

N2 - We prove that if D is a domain in ℂ, α > 1 and C > 0, then the family F of functions f meromorphic in D such that {pipe}f′(z)/1+{pipe}f(z){pipe}α > C for every z ∈ D is normal in D. For α = 1, the same assumptions imply quasi-normality but not necessarily normality.

AB - We prove that if D is a domain in ℂ, α > 1 and C > 0, then the family F of functions f meromorphic in D such that {pipe}f′(z)/1+{pipe}f(z){pipe}α > C for every z ∈ D is normal in D. For α = 1, the same assumptions imply quasi-normality but not necessarily normality.

KW - Normal family

KW - differential inequality

KW - quasi-normal family

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DO - 10.1007/s10114-014-2542-8

M3 - Article

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SP - 277

EP - 282

JO - Acta Mathematica Sinica, English Series

JF - Acta Mathematica Sinica, English Series

IS - 2

ER -

Differential inequalities, normality and quasi-normality

Abstract

Keywords

ASJC Scopus subject areas

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