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

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

7 Scopus citations

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

Bibliographical 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)

Funding

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)

Funders	Funder number
Tianyuan Special Funds of the National Natural Science Foundation of China	11226095
National Natural Science Foundation of China	11071074
Israel Science Foundation	395/07
National Outstanding Youth Foundation of China	slg10015

Keywords

Normal family
differential inequality
quasi-normal family

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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.",

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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.

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Differential inequalities, normality and quasi-normality

Abstract

Bibliographical note

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Keywords

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