Differential Inequalities, Normality and Quasi-normality
We prove that ifD is a domain in C,α 〉 1 and C 〉 0,then the family F of functions f meromorphic in D such that ｜f′（z）｜/1 ＋ ｜f（z）｜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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| Published in | Acta mathematica Sinica. English series Vol. 30; no. 2; pp. 277 - 282 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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Berlin/Heidelberg
Springer Berlin Heidelberg
01.02.2014
Springer Nature B.V |
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| Abstract | We prove that ifD is a domain in C,α 〉 1 and C 〉 0,then the family F of functions f meromorphic in D such that ｜f′（z）｜/1 ＋ ｜f（z）｜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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| AbstractList | (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).We prove that if D is a domain in , alpha > 1 and C > 0, then the family F of functions f meromorphic in D such that ... is normal in D. For alpha = 1, the same assumptions imply quasi-normality but not necessarily normality. We prove that if D is a domain in , [alpha] > 1 and C > 0, then the family F of functions f meromorphic in D such that ... is normal in D. For [alpha] = 1, the same assumptions imply quasi-normality but not necessarily normality. [PUBLICATION 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 is normal in D . For α = 1, the same assumptions imply quasi-normality but not necessarily normality. We prove that ifD is a domain in C,α 〉 1 and C 〉 0,then the family F of functions f meromorphic in D such that ｜f′（z）｜/1 ＋ ｜f（z）｜α 〉 C for every z ∈ D is normal in D.For α = 1,the same assumptions imply quasi-normality but not necessarily normality. |
| Author | Xiao Jun LIU Shahar NEVO Xue Cheng PANG |
| AuthorAffiliation | Department of Mathematics, University of Shanghai for Science and Technology, Shanghai 200093, P. R. China Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel Department of Mathematics, East China Normal University, Shanghai 200241, P. R. China |
| Author_xml | – sequence: 1 givenname: Xiao Jun surname: Liu fullname: Liu, Xiao Jun email: Xiaojunliu2007@hotmail.com organization: Department of Mathematics, University of Shanghai for Science and Technology – sequence: 2 givenname: Shahar surname: Nevo fullname: Nevo, Shahar organization: Department of Mathematics, Bar-Ilan University – sequence: 3 givenname: Xue Cheng surname: Pang fullname: Pang, Xue Cheng organization: Department of Mathematics, East China Normal University |
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| Cites_doi | 10.1007/978-1-4612-0907-2 10.1007/s11854-012-0017-3 10.1007/s11854-012-0016-4 10.1112/S002460939900644X 10.1090/S0273-0979-98-00755-1 10.1142/1904 |
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| Copyright | Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2014 |
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| DOI | 10.1007/s10114-014-2542-8 |
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| Notes | 11-2039/O1 We prove that ifD is a domain in C,α 〉 1 and C 〉 0,then the family F of functions f meromorphic in D such that ｜f′（z）｜/1 ＋ ｜f（z）｜α 〉 C for every z ∈ D is normal in D.For α = 1,the same assumptions imply quasi-normality but not necessarily normality. Normal family, quasi-normal family, differential inequality SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-2 content type line 23 |
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| References | SchwickWOn a normality criterion of H. L. RoydenNew Zealand J. Math.19942391920824.300221279130 SteinmetzNNormal families and linear differential equationJ. Anal. Math.201211712913210.1007/s11854-012-0017-3061869512944093 ChuangC TNormal Families of Meromorphic Functions1993SingaporeWorld Scientific10.1142/19040878.30026 HinkkanenANormal families and Ahlfor’s five island theoremNew Zealand J. Math.19932239410796.300291244021 PangX CZalcmanLNormal families and shared valuesBull. London Math. Soc.20003232533110.1112/S002460939900644X1030.300311750485 RoydenH LA criterion for the normality of a family of meromorphic functionsAnn. Acad. Sci. Fenn. Math.1985104995000588.30034802513 ZalcmanLNormal families: new perspectivesBull. Amer. Math. Soc. (N.S.)19983521523010.1090/S0273-0979-98-00755-11037.300211624862 SchiffJNormal Families1993New YorkSpringer10.1007/978-1-4612-0907-20770.30002 GuY XA criterion for normality of families of meromorphic functionsSci. Sinica (special issue)19791267274 GrahlJNevoSSpherical derivatives and normal familiesJ. Anal. Math.201211711912810.1007/s11854-012-0016-4061869502944092 A Hinkkanen (2542_CR4) 1993; 22 L Zalcman (2542_CR10) 1998; 35 J Grahl (2542_CR2) 2012; 117 Y X Gu (2542_CR3) 1979; 1 J Schiff (2542_CR7) 1993 C T Chuang (2542_CR1) 1993 H L Royden (2542_CR6) 1985; 10 W Schwick (2542_CR8) 1994; 23 N Steinmetz (2542_CR9) 2012; 117 X C Pang (2542_CR5) 2000; 32 |
| References_xml | – reference: SchiffJNormal Families1993New YorkSpringer10.1007/978-1-4612-0907-20770.30002 – reference: ChuangC TNormal Families of Meromorphic Functions1993SingaporeWorld Scientific10.1142/19040878.30026 – reference: SteinmetzNNormal families and linear differential equationJ. Anal. Math.201211712913210.1007/s11854-012-0017-3061869512944093 – reference: SchwickWOn a normality criterion of H. L. RoydenNew Zealand J. Math.19942391920824.300221279130 – reference: GrahlJNevoSSpherical derivatives and normal familiesJ. Anal. Math.201211711912810.1007/s11854-012-0016-4061869502944092 – reference: HinkkanenANormal families and Ahlfor’s five island theoremNew Zealand J. Math.19932239410796.300291244021 – reference: ZalcmanLNormal families: new perspectivesBull. Amer. Math. Soc. (N.S.)19983521523010.1090/S0273-0979-98-00755-11037.300211624862 – reference: PangX CZalcmanLNormal families and shared valuesBull. London Math. Soc.20003232533110.1112/S002460939900644X1030.300311750485 – reference: RoydenH LA criterion for the normality of a family of meromorphic functionsAnn. Acad. Sci. Fenn. Math.1985104995000588.30034802513 – reference: GuY XA criterion for normality of families of meromorphic functionsSci. Sinica (special issue)19791267274 – volume: 1 start-page: 267 year: 1979 ident: 2542_CR3 publication-title: Sci. Sinica (special issue) – volume-title: Normal Families year: 1993 ident: 2542_CR7 doi: 10.1007/978-1-4612-0907-2 – volume: 117 start-page: 129 year: 2012 ident: 2542_CR9 publication-title: J. Anal. Math. doi: 10.1007/s11854-012-0017-3 – volume: 23 start-page: 91 year: 1994 ident: 2542_CR8 publication-title: New Zealand J. Math. – volume: 117 start-page: 119 year: 2012 ident: 2542_CR2 publication-title: J. Anal. Math. doi: 10.1007/s11854-012-0016-4 – volume: 22 start-page: 39 year: 1993 ident: 2542_CR4 publication-title: New Zealand J. Math. – volume: 32 start-page: 325 year: 2000 ident: 2542_CR5 publication-title: Bull. London Math. Soc. doi: 10.1112/S002460939900644X – volume: 10 start-page: 499 year: 1985 ident: 2542_CR6 publication-title: Ann. Acad. Sci. Fenn. Math. – volume: 35 start-page: 215 year: 1998 ident: 2542_CR10 publication-title: Bull. Amer. Math. Soc. (N.S.) doi: 10.1090/S0273-0979-98-00755-1 – volume-title: Normal Families of Meromorphic Functions year: 1993 ident: 2542_CR1 doi: 10.1142/1904 |
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| Snippet | We prove that ifD is a domain in C,α 〉 1 and C 〉 0,then the family F of functions f meromorphic in D such that ｜f′（z）｜/1 ＋ ｜f（z）｜α 〉 C for every z ∈ D is... 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 is normal in D . For α = 1, the same... We prove that if D is a domain in , [alpha] > 1 and C > 0, then the family F of functions f meromorphic in D such that ... is normal in D. For [alpha] = 1, the... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).We prove that if D is a domain in , alpha > 1 and C > 0, then the family F of... |
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| Title | Differential Inequalities, Normality and Quasi-normality |
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