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.

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.

Normality and shared values concerning differential polynomial

2011 ◽  
Vol 18 (2) ◽  
pp. 299-306
Author(s):  
Chunlin L. Lei ◽  
Degui G. Yang ◽  
Cuiping P. Zeng
Keyword(s):  
Positive Integer ◽  
Meromorphic Functions ◽  
Differential Polynomial ◽  
Shared Values ◽  
Complex Numbers

Abstract Let ℱ be a family of meromorphic functions in a domain D; let k ≥ 2 be a positive integer; and let a, b and c be complex numbers such that b ≠ 0 and a ≠ c. If, for each ƒ ∈ ℱ, all zeros of ƒ have multiplicity at least k, ƒ(z) = a ⇔ D(ƒ) = b, and D(ƒ) = 0 ⇒ ƒ(z) = c, where D(ƒ) is the differential polynomial of ƒ(z), then ℱ is normal in D.

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 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.

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.

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 normality and shared values

2004 ◽  
Vol 76 (1) ◽  
pp. 141-150 ◽  
Author(s):  
Mingliang Fang ◽  
Lawrence Zalcman
Keyword(s):  
Positive Integer ◽  
Meromorphic Functions ◽  
Shared Values ◽  
Nonzero Constant

AbstractLet k be a positive integer and b a nonzero constant. Suppose that F is a family of meromorphic functions in a domain D. If each function f ∈ F has only zeros of multiplicity at least k + 2 and for any two functions f, g ∈ F, f and g share 0 in D and f(k) and g(k) share b in D, then F is normal in D. The case f ≠ 0, f(k) ≠ b is a celebrated result of Gu.

scholarly journals Value distribution of meromorphic functions concerning rational functions and differences

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mingliang Fang ◽  
Degui Yang ◽  
Dan Liu
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Rational Function ◽  
Finite Order ◽  
Meromorphic Functions ◽  
Rational Functions ◽  
Value Distribution ◽  
Exponent Of Convergence ◽  
Nonzero Constant ◽  
Zero Points

AbstractLet c be a nonzero constant and n a positive integer, let f be a transcendental meromorphic function of finite order, and let R be a nonconstant rational function. Under some conditions, we study the relationships between the exponent of convergence of zero points of $f-R$ f − R , its shift $f(z+nc)$ f ( z + n c ) and the differences $\Delta _{c}^{n} f$ Δ c n f .

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