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 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.

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 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 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 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 .

Sign in / Sign up

Export Citation Format

Share Document