Differential inequalities and a marty-type criterion for Quasi-normality

Jürgen Grahl; Tomer Manket; Shahar Nevo

doi:10.1017/S1446788717000180

Differential inequalities and a marty-type criterion for Quasi-normality

Jürgen Grahl, Tomer Manket, Shahar Nevo

Department of Mathematics

University of Würzburg

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

2 Scopus citations

Abstract

We show that the family of all holomorphic functions in a f domain D satisfying |f^(k)|/1+|f| (z) ≤ C for all z ∞ D (where k is a natural number and C> 0) is quasi-normal. Furthermore, we give a general counterexample to show that for α> 1 and k≥2 the condition |f^(k)|/1+|f|^α (z) ≤ C for all z ∞ D does not imply quasi-normality.

Original language	English
Pages (from-to)	34-45
Number of pages	12
Journal	Journal of the Australian Mathematical Society
Volume	105
Issue number	1
DOIs	https://doi.org/10.1017/S1446788717000180
State	Published - 1 Aug 2018

Bibliographical note

Publisher Copyright:
© 2018 Australian Mathematical Publishing Association Inc.

Keywords

Marty's theorem
differential inequalities
normal families
quasi-normal families

Access to Document

10.1017/S1446788717000180

Cite this

@article{e65c3a87e8b741adae7769624af1d3a3,

title = "Differential inequalities and a marty-type criterion for Quasi-normality",

abstract = "We show that the family of all holomorphic functions in a f domain D satisfying |f(k)|/1+|f| (z) ≤ C for all z ∞ D (where k is a natural number and C> 0) is quasi-normal. Furthermore, we give a general counterexample to show that for α> 1 and k≥2 the condition |f(k)|/1+|f|α (z) ≤ C for all z ∞ D does not imply quasi-normality.",

keywords = "Marty's theorem, differential inequalities, normal families, quasi-normal families",

author = "J{\"u}rgen Grahl and Tomer Manket and Shahar Nevo",

note = "Publisher Copyright: {\textcopyright} 2018 Australian Mathematical Publishing Association Inc.",

year = "2018",

month = aug,

day = "1",

doi = "10.1017/S1446788717000180",

language = "אנגלית",

volume = "105",

pages = "34--45",

journal = "Journal of the Australian Mathematical Society",

issn = "1446-7887",

publisher = "Cambridge University Press",

number = "1",

}

TY - JOUR

T1 - Differential inequalities and a marty-type criterion for Quasi-normality

AU - Grahl, Jürgen

AU - Manket, Tomer

AU - Nevo, Shahar

PY - 2018/8/1

Y1 - 2018/8/1

N2 - We show that the family of all holomorphic functions in a f domain D satisfying |f(k)|/1+|f| (z) ≤ C for all z ∞ D (where k is a natural number and C> 0) is quasi-normal. Furthermore, we give a general counterexample to show that for α> 1 and k≥2 the condition |f(k)|/1+|f|α (z) ≤ C for all z ∞ D does not imply quasi-normality.

AB - We show that the family of all holomorphic functions in a f domain D satisfying |f(k)|/1+|f| (z) ≤ C for all z ∞ D (where k is a natural number and C> 0) is quasi-normal. Furthermore, we give a general counterexample to show that for α> 1 and k≥2 the condition |f(k)|/1+|f|α (z) ≤ C for all z ∞ D does not imply quasi-normality.

KW - Marty's theorem

KW - differential inequalities

KW - normal families

KW - quasi-normal families

UR - http://www.scopus.com/inward/record.url?scp=85048803868&partnerID=8YFLogxK

U2 - 10.1017/S1446788717000180

DO - 10.1017/S1446788717000180

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85048803868

SN - 1446-7887

VL - 105

SP - 34

EP - 45

JO - Journal of the Australian Mathematical Society

JF - Journal of the Australian Mathematical Society

IS - 1

ER -

Differential inequalities and a marty-type criterion for Quasi-normality

Abstract

Bibliographical note

Keywords

Access to Document

Other files and links

Fingerprint

Cite this