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

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.

2012 ◽  
Vol 29 (1) ◽  
pp. 151-158 ◽  
Author(s):  
Xiao Jun Liu ◽  
San Hua Li ◽  
Xue Cheng Pang

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Xin-Li Wang ◽  
Ning Cui

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.


2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Wei Chen ◽  
Honggen Tian ◽  
Yingying Zhang ◽  
Wenjun Yuan

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.


Author(s):  
Chengxiong Sun

AbstractLet $$n \ge 4$$ n ≥ 4 be a positive integer, $$\mathcal {F}$$ F be a family of meromorphic functions in D and let $$a(z)(\not \equiv 0), b(z)$$ a ( z ) ( ≢ 0 ) , b ( z ) be two holomorphic functions in D. If, for any function $$f \in \mathcal { F}$$ f ∈ F , (1)$$f(z) \ne \infty $$ f ( z ) ≠ ∞ when $$a(z)=0$$ a ( z ) = 0 , (2) $$f'(z)-a(z)f^{n}(z)-b(z)$$ f ′ ( z ) - a ( z ) f n ( z ) - b ( z ) has at most one zero in D, then $$\mathcal {F}$$ F is normal in D.


2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Wei Chen ◽  
Yingying Zhang ◽  
Jiwen Zeng ◽  
Honggen Tian

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.


2011 ◽  
Vol 18 (2) ◽  
pp. 299-306
Author(s):  
Chunlin L. Lei ◽  
Degui G. Yang ◽  
Cuiping P. Zeng

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.


2004 ◽  
Vol 76 (1) ◽  
pp. 141-150 ◽  
Author(s):  
Mingliang Fang ◽  
Lawrence Zalcman

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.


2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
Wenjun Yuan ◽  
Bing Zhu ◽  
Jianming Lin

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.


2007 ◽  
Vol 8 (2) ◽  
pp. 483-491 ◽  
Author(s):  
Xuecheng Pang ◽  
Shahar Nevo ◽  
Lawrence Zalcman

2013 ◽  
Vol 120 (1) ◽  
pp. 131-149
Author(s):  
Thierry Meyrath ◽  
Jürgen Müller

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