A normality criterion for meromorphic functions having multiple zeros

2014 ◽  
Vol 110 (3) ◽  
pp. 283-294 ◽  
Author(s):  
Shanpeng Zeng ◽  
Indrajit Lahiri
Keyword(s):  
Meromorphic Functions ◽  
Multiple Zeros

On the value distribution of f2f(k)

2005 ◽  
Vol 78 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Xiaojun Huang ◽  
Yongxing Gu
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Value Distribution ◽  
Multiple Zeros ◽  
Transcendental Meromorphic Function

AbstractIn this paper, we prove that for a transcendental meromorphic function f(z) on the complex plane, the inequality T(r, f) < 6N (r, 1/(f2 f(k)−1)) + S(r, f) holds, where k is a positive integer. Moreover, we prove the following normality criterion: Let ℱ be a family of meromorphic functions on a domain D and let k be a positive integer. If for each ℱ ∈ ℱ, all zeros of ℱ are of multiplicity at least k, and f2 f(k) ≠ 1 for z ∈ D, then ℱ is normal in the domain D. At the same time we also show that the condition on multiple zeros of f in the normality criterion is necessary.

scholarly journals Derivatives of Meromorphic Functions with Multiple Zeros and Small Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Pai Yang ◽  
Peiyan Niu
Keyword(s):  
Meromorphic Function ◽  
Rational Function ◽  
Elliptic Function ◽  
Meromorphic Functions ◽  
Infinitely Many Solutions ◽  
Multiple Zeros ◽  
Small Functions ◽  
Derivatives Of

Letfzbe a meromorphic function inℂ, and letαz=Rzhz≢0, wherehzis a nonconstant elliptic function andRzis a rational function. Suppose that all zeros offzare multiple except finitely many andTr,α=oTr,fasr→∞. Thenf'z=αzhas infinitely many solutions.

Exceptional functions and normal families of holomorphic functions with multiple zeros

2011 ◽  
Vol 18 (1) ◽  
pp. 31-38
Author(s):  
Jun-Fan Chen
Keyword(s):  
Positive Integer ◽  
Meromorphic Functions ◽  
Normal Family ◽  
Holomorphic Functions ◽  
Normal Families ◽  
Multiple Zeros ◽  
Zeros Of Functions

Abstract Let k be a positive integer, and let ℱ be a family of functions holomorphic on a domain D in C, all of whose zeros are of multiplicity at least k + 1. Let h be a function meromorphic on D, h ≢ 0, ∞. Suppose that for each ƒ ∈ ℱ, ƒ(k)(z) ≠ h(z) for z ∈ D. Then ℱ is a normal family on D. The condition that the zeros of functions in ℱ are of multiplicity at least k + 1 cannot be weakened, and the corresponding result for families of meromorphic functions is no longer true.

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].

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