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.

NORMALITY CRITERIA FOR FAMILIES OF MEROMORPHIC FUNCTIONS WITH SHARED VALUES

2015 ◽  
Vol 1 (2) ◽  
pp. 45-48
Author(s):  
Shyamali Dewan
Keyword(s):  
Meromorphic Functions ◽  
Normal Family ◽  
Shared Values ◽  
Multiple Zeros ◽  
Positive Integers ◽  
Multiple Poles

In this paper we have discussed normality criteria of a family of meromorphic functions. We have studied whether a family of meromorphic functions $\mathcal{F}$ is normal in $D$ if for a normal family $G$ and for each function $f\in \mathcal{F} $ there exists $g\in G$ such that $(f^{(k)})^n = a_i$ implies $(g^{(k)})^n = a_i$, $i=1,2,\ldots$ for two distinct non zero constants $a_i$ and $n (\ge 2)$, $k$ being positive integers. In this approach we have considered the functions with multiple zeros and multiple poles. We also have proved another result which improves the result of Yuan et al. [1].

Picard values and normal families of meromorphic functions

2009 ◽  
Vol 139 (5) ◽  
pp. 1091-1099 ◽  
Author(s):  
Yan Xu ◽  
Fengqin Wu ◽  
Liangwen Liao
Keyword(s):  
Meromorphic Function ◽  
Complex Plane ◽  
Complex Number ◽  
Meromorphic Functions ◽  
Normal Families ◽  
Finite Complex ◽  
Transcendental Meromorphic Function ◽  
Positive Integers ◽  
Normal Criterion

Let f be a transcendental meromorphic function on the complex plane ℂ, let a be a non-zero finite complex number and let n and k be two positive integers. In this paper, we prove that if n≥k+1, then $\smash{f+a(f^{(k)})^n}$ assumes each value b∈ℂ infinitely often. Also, the related normal criterion for families of meromorphic functions is given. Our results generalize the related results of Fang and Zalcman.

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 Criterion Concerning Shared Values

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Wei Chen ◽  
Yingying Zhang ◽  
Jiwen Zeng ◽  
Honggen Tian
Keyword(s):  
Meromorphic Functions ◽  
Shared Values ◽  
Complex Numbers ◽  
Normal Criterion

We study normal criterion of meromorphic functions shared values, we obtain the following. LetFbe a family of meromorphic functions in a domainD, such that functionf∈Fhas zeros of multiplicity at least 2, there exists nonzero complex numbersbf,cfdepending onfsatisfying(i)  bf/cfis a constant;  (ii)min {σ(0,bf),σ(0,cf),σ(bf,cf)≥m}for somem>0;  (iii)  (1/cfk-1)(f′)k(z)+f(z)≠bfk/cfk-1or(1/cfk-1)(f′)k(z)+f(z)=bfk/cfk-1⇒f(z)=bf, thenFis normal. These results improve some earlier previous results.

scholarly journals Normal criterion and shared values by derivatives of meromorphic functions

2014 ◽  
Vol 45 (2) ◽  
pp. 109-117
Author(s):  
Qian Lu ◽  
Qilong Liao
Keyword(s):  
Positive Integer ◽  
Meromorphic Functions ◽  
Plane Domain ◽  
Shared Values ◽  
Derivatives Of ◽  
Normal Criterion

Let $\mathscr{F}$ be a family of meromorphic functions in a plane domain $D$. If for every function $f\in\mathscr{F}$, all of whose zeros have,at least,multiplicity $l$ and poles have, at least,multiplicity $p$, and for each pair functions $f$ and $g$ in $\mathscr{F}$, $f^{(k)}$ and $g^{(k)}$ share 1 in $D$, where $k,l,$ and $p$ are three positive integer satisfying $\frac{k+1}{l}+\frac{1}{p}\leq 1$, then $\mathscr{F}$ is normal.

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 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.

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