scholarly journals A Note on Normal Families of Meromorphic Functions Concerning Shared Values

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Yuan Wenjun ◽  
Wei Jinjin ◽  
Lin Jianming
Keyword(s):  
Meromorphic Functions ◽  
Shared Values ◽  
Normal Families ◽  
Complex Numbers ◽  
Finite Complex

We study the normality of families of meromorphic functions related to a Hayman conjecture. We consider whether a family of meromorphic functionsℱis normal inDif, for every pair of functionsfandginℱ,f′−afnandg′−agnshare the valuebforn=1,2, and 3, whereaandb≠0are two finite complex numbers. Some examples show that the conditions in our results are the best possible.

scholarly journals Some Normal Criteria about Shared Values with Their Multiplicity Zeros

2010 ◽  
Vol 2010 ◽  
pp. 1-14 ◽  
Author(s):  
Jianming Qi ◽  
Taiying Zhu
Keyword(s):  
Meromorphic Functions ◽  
Shared Values ◽  
Complex Numbers ◽  
Finite Complex

LetFbe a family of meromorphic functions in the domainD, all of whose zeros are multiple. Letn  (n≥2)be an integer and leta,bbe two nonzero finite complex numbers. Iff+a(f')nandg+a(g')nsharebinDfor every pair of functionsf,g∈F, thenFis normal inD.

scholarly journals Normal Criteria of Function Families Concerning Shared Values

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
Wenjun Yuan ◽  
Bing Zhu ◽  
Jianming Lin
Keyword(s):  
Positive Integer ◽  
Meromorphic Functions ◽  
Shared Values ◽  
Complex Numbers ◽  
Finite Complex

We study the normality of families of meromorphic functions concerning shared values. We consider whether a family of meromorphic functions ℱ is normal inD, if, for every pair of functionsfandgin ℱ,f′−af−nandg′−ag−nshare the valueb, whereaandbare two finite complex numbers such thata≠0,nis a positive integer. Some examples show that the conditions in our results are best possible.

scholarly journals NORMALITY AND SHARED SETS

2009 ◽  
Vol 86 (3) ◽  
pp. 339-354 ◽  
Author(s):  
MINGLIANG FANG ◽  
LAWRENCE ZALCMAN
Keyword(s):  
Positive Integer ◽  
Meromorphic Functions ◽  
Complex Numbers ◽  
Finite Complex ◽  
Shared Sets ◽  
Zeros Of Functions

AbstractLet ℱ be a family of meromorphic functions defined in D, all of whose zeros have multiplicity at least k+1. Let a and b be distinct finite complex numbers, and let k be a positive integer. If, for each pair of functions f and g in ℱ, f(k) and g(k) share the set S={a,b}, then ℱ is normal in D. The condition that the zeros of functions in ℱ have multiplicity at least k+1 cannot be weakened.

Normal families of meromorphic functions concerning shared values

2013 ◽  
Vol 58 (1) ◽  
pp. 113-121 ◽  
Author(s):  
Jian-Jun Ding ◽  
Li-Wei Ding ◽  
Wen-Jun Yuan
Keyword(s):  
Meromorphic Functions ◽  
Shared Values ◽  
Normal Families

scholarly journals Normal Families concerning Polynomials and Shared Values

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Xin-Li Wang ◽  
Ning Cui
Keyword(s):  
Positive Integer ◽  
Meromorphic Functions ◽  
Shared Values ◽  
Normal Families

We study the problem of normal families of meromorphic functions concerning polynomials and shared values. We prove that a family ℱ of meromorphic functions in a domain D is normal if, for each function f∈ℱ, Pfzfkz=a⇔fkz=b, where P is a polynomial with the origin as zero, k is a positive integer, and a ≠0, b are two finite constants.

Normal families of meromorphic functions with shared values

2016 ◽  
Vol 36 (1) ◽  
pp. 87-93 ◽  
Author(s):  
Wei CHEN ◽  
Honggen TIAN ◽  
Peichu HU
Keyword(s):  
Meromorphic Functions ◽  
Shared Values ◽  
Normal Families

scholarly journals Normal Family of Meromorphic Functions concerning Shared Values

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Wei Chen ◽  
Honggen Tian ◽  
Yingying Zhang ◽  
Wenjun Yuan
Keyword(s):  
Meromorphic Functions ◽  
Normal Family ◽  
Shared Values ◽  
Finite Complex ◽  
Related Theorem ◽  
Positive Integers ◽  
Normal Criterion

We obtain a normal criterion of meromorphic functions concerning, shared values. Let ℱ be a family of meromorphic functions in a domain D and let k,n≥k+2 be positive integers. Let a≠0,b be two finite complex constants. If, for each f∈ℱ, all zeros of f have multiplicity at least k+1 and f+a(f(k))n and g+a(g(k))n share b in D for every pair of functions f,g∈ℱ, then ℱ is normal in D. This result generalizes the related theorem according to Xu et al. and Qi et al., respectively. There is a gap in the proofs of Lemma 3 by Wang (2012) and Theorem 1 by Zhang (2008), respectively. They did not consider the case of f(z) being zerofree. We will fill the gap in this paper.

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