Normal criteria on the family of meromorphic functions with shared set
type: Item
instance of: scholarly article
title: Normal criteria on the family of meromorphic functions with shared set
zbMATH Open document ID: 1278.30034
author: Daochun Sun
author: Huifang Liu
published in: Israel Journal of Mathematics
publication date: 2012-11-19T00:00:00Z
review text: Let \(\mathcal{F}\) be a family of meromorphic functions in a domain \(D\) of the extended complex plane \(\mathbb{C}^1\) and let \(S\subset V\) be a non-empty finite point set on the relevant Riemann sphere of unit diameter. Using some geometric methods and the theory of covering surfaces, the authors prove the following results about normality of \(\mathcal{F}\) with some corollaries: {Theorem 1.1} (Area criteria): Suppose that \(\mathcal{F}\) is a family of meromorphic functions in a domain \(D\subset \mathbb{C}^1\). Then the following conditions are equivalent: (i) \(\mathcal{F}\) is normal in \(D\). (ii) \(\mathcal{F}\) is equicontinuous in \(D\). (iii) For any \(\varepsilon \in (0, \pi)\) and any \(z^* \in D\), there exists a positive number \(r=r(z^*)>0\) such that, for any \(f \in \mathcal{F}\), the area of the image set \(f(\{z:|z-z^*|<r\})\) is smaller than \(\varepsilon\). {Theorem 1.2}: Suppose that \(\mathcal{F}\) is a family of meromorphic functions in a domain \(D\subset \mathbb{C}^1\) and \(S\subset V\) is a quasi-shared of \(\mathcal{F}\). If \(S\) contains at least three points, then \(\mathcal{F}\) is normal in \(D\). {Theorem 1.3}: Suppose that \(\mathcal{F}\) is a family of meromorphic functions in a domain \(D\subset \mathbb{C}^1\) and \(S\subset V\) is a shared of \(\mathcal{F}\). If \(S\) contains at least three points, then \(\mathcal{F}\) is normal in \(D\). {Theorem 1.4}: Suppose that \(\mathcal{F}\) is a family of meromorphic functions in the domain \(D\subset \mathbb{C}^1\). If \(\mathcal{F}\) has three distinct shared values, then \(\mathcal{F}\) is normal in \(D\). {Corollary 1.} Suppose that \(\mathcal{F}\) is a family of meromorphic functions in a domain \(D\subset \mathbb{C}^1\). If there exist three distinct points \(a_1, a_2, a_3 \in V\) such that, for any given \(f \in \mathcal{F}\), \(f\) omits \(a_1, a_2, a_3\) in \(D\), then \(\mathcal{F}\) is normal in \(D\).
Mathematics Subject Classification ID: 30D45
zbMATH DE Number: 6108364
zbMATH Keywords: extended complex plane
zbMATH Keywords: family of meromorphic functions defined on \(D\subset \mathbb{C}^1\)
zbMATH Keywords: normality properties
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
full work available at URL: s11856-011-0073-x
OpenAlex ID: W1999039327
cites work: Shared values and normal families of meromorphic functions
cites work: Normal families concerning shared values
cites work: Normal families of holomorphic functions and shared values
cites work: Normal families and shared values of meromorphic functions
cites work: Normal families and shared values of holomorphic functions
cites work: Q3148718
cites work: Q3259806
cites work: Normal families of meromorphic functions concerning shared values
DOI: 10.1007/S11856-011-0073-X
description: scientific article; zbMATH DE number 6108364