On Uniqueness of Meromorphic Functions with Shared Values in Some Angular Domains

2004 ◽  
Vol 47 (1) ◽  
pp. 152-160 ◽  
Author(s):  
Zheng Jian-Hua
Keyword(s):  
Meromorphic Function ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Shared Values ◽  
Transcendental Meromorphic Function ◽  
Angular Domains

AbstractIn this paper we investigate the uniqueness of transcendental meromorphic function dealing with the shared values in some angular domains instead of the whole complex plane.

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.

The Distribution of Finite Values of Meromorphic Functions with Deficient Poles

2008 ◽  
Vol 51 (3) ◽  
pp. 697-709
Author(s):  
G. F. Kendall
Keyword(s):  
Meromorphic Function ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Transcendental Meromorphic Function

AbstractA result is presented giving conditions on a set of open discs in the complex plane that ensure that a transcendental meromorphic function with Nevanlinna deficient poles omits at most one finite value outside the set of discs. This improves a previous result of Langley, and goes some way towards closing a gap between Langley's result and a theorem of Toppila in which the omitted values considered may include ∞

scholarly journals Value distribution of some differential monomials

Filomat ◽  
2020 ◽  
Vol 34 (13) ◽  
pp. 4287-4295 ◽  
Author(s):  
Bikash Chakraborty ◽  
Sudip Saha ◽  
Amit Pal ◽  
Jayanta Kamila
Keyword(s):  
Meromorphic Function ◽  
Characteristic Function ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Quantitative Estimation ◽  
Differential Polynomial ◽  
Value Distribution ◽  
Transcendental Meromorphic Function

Let f be a transcendental meromorphic function defined in the complex plane C and k ? N. We consider the value distribution of the differential polynomial fq0(f(k))qk, where q0(?2), qk(?1) are integers. We obtain a quantitative estimation of the characteristic function T(r,f) in terms of N?(r, 1/fq0(f(k))qk-1). Our result generalizes the results obtained by Xu et al. (Math. Inequal. Appl., Vol. 14, PP. 93-100, 2011); Karmakar and Sahoo (Results Math., Vol. 73, 2018) for a particular class of transcendental meromorphic functions.

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 Fixed Points of Difference Operator of Meromorphic Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Zhaojun Wu ◽  
Hongyan Xu
Keyword(s):  
Meromorphic Function ◽  
Fixed Points ◽  
Meromorphic Functions ◽  
Difference Operator ◽  
Transcendental Meromorphic Function ◽  
Exact Difference ◽  
Borel Exceptional Values ◽  
Nevanlinna Deficiency

Letfbe a transcendental meromorphic function of order less than one. The authors prove that the exact differenceΔf=fz+1-fzhas infinitely many fixed points, ifa∈ℂand∞are Borel exceptional values (or Nevanlinna deficiency values) off. These results extend the related results obtained by Chen and Shon.

DIMENSIONS OF JULIA SETS OF HYPERBOLIC MEROMORPHIC FUNCTIONS

2001 ◽  
Vol 33 (6) ◽  
pp. 689-694 ◽  
Author(s):  
GWYNETH M. STALLARD
Keyword(s):  
Hausdorff Dimension ◽  
Meromorphic Function ◽  
Rational Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Transcendental Meromorphic Function ◽  
Box Dimensions

It is known that, if f is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set, J(f), are equal. In this paper it is shown that, for a hyperbolic transcendental meromorphic function f, the packing and upper box dimensions of J(f) are equal, but can be strictly greater than the Hausdorff dimension of J(f).

scholarly journals Continuity of Cima and Rung's extension and non normal meromorphic functions

1979 ◽  
Vol 20 (1) ◽  
pp. 139-143
Author(s):  
Douglas M. Campbell
Keyword(s):  
Meromorphic Function ◽  
Complex Plane ◽  
Open Problem ◽  
Meromorphic Functions ◽  
Continuous Extension ◽  
Entire Complex ◽  
Long Time ◽  
Limit Points

A function meromorphic in |z| < 1 is constructed such that on every curve in |z| < 1 which goes to |z| = 1 the set of limit points of the function is the entire complex plane. This example is used to prove the existence of non-normal meromorphic functions in |z| < 1 which have continuous set valued extensions. Cima and Rung had introduced a set valued extension for meromorphic functions and proved that all normal meromorphic functions have a continuous extension while all functions with a continuous extension have the Lindelöf property. For a long time it was thought that this might characterize normal meromorphic functions. This paper proves that it is not possible to determine the normality of a meromorphic function by the continuity of Cima and Rung's set valued extension. The paper closes with the open problem: do there exist non-normal analytic functions for which Cima and Rung's set valued extension is continuous?

ON THE ESCAPING SET OF MEROMORPHIC FUNCTIONS WITH DIRECT TRACTS

2015 ◽  
Vol 92 (1) ◽  
pp. 68-76
Author(s):  
ZUXING XUAN ◽  
JIANHUA ZHENG
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Transcendental Meromorphic Function

Let $f$ be a transcendental meromorphic function with at least one direct tract. In this note, we investigate the structure of the escaping set which is in the same direct tract. We also give a theorem about the slow escaping set.

scholarly journals On the common right factors of meromorphic functions

1997 ◽  
Vol 55 (3) ◽  
pp. 395-403 ◽  
Author(s):  
Tuen-Wai Ng ◽  
Chung-Chun Yang
Keyword(s):  
Entire Function ◽  
Meromorphic Function ◽  
Rational Function ◽  
Finite Order ◽  
Meromorphic Functions ◽  
Transcendental Meromorphic Function ◽  
The Common ◽  
Periodic Entire Function

In this paper, common right factors (in the sense of composition) of p1 + p2F and p3 + p4F are investigated. Here, F is a transcendental meromorphic function and pi's are non-zero polynomials. Moreover, we also prove that the quotient (p1 + p2F)/(p3 + p4F) is pseudo-prime under some restrictions on F and the pi's. As an application of our results, we have proved that R (z) H (z)is pseudo-prime for any nonconstant rational function R (z) and finite order periodic entire function H (z).

scholarly journals The partially shared values and small functions for meromorphic functions in a k-punctured complex plane

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hong Yan Xu ◽  
Yong Ming Li ◽  
Shan Liu
Keyword(s):  
Complex Plane ◽  
Meromorphic Functions ◽  
Shared Values ◽  
Small Functions

Abstract The main aim of this article is to discuss the uniqueness of meromorphic functions partially sharing some values and small functions in a k-punctured complex plane Ω. We proved the following: Let $f_{1},f_{2}$f1,f2 be two admissible meromorphic functions in Ω and $\alpha _{j}\ (j=1,2,\ldots ,l)$αj(j=1,2,…,l) be $l(\geq 5)$l(≥5) distinct small functions with respect to f and g. If $\widetilde{E}(\alpha _{j},\varOmega ,f_{1})\subseteq \widetilde{E}(\alpha _{j},\varOmega , f_{2})\ (j=1,2,\ldots ,l)$E˜(αj,Ω,f1)⊆E˜(αj,Ω,f2)(j=1,2,…,l) and $$ \liminf_{r\rightarrow +\infty }\frac{\sum_{j=1}^{l}\overline{N} _{0} (r,\frac{1}{f_{1}-\alpha _{j}} )}{\sum_{j=1} ^{l}\overline{N}_{0} (r,\frac{1}{f_{2}-\alpha _{j}} )}> \frac{5}{2l-5}, $$lim infr→+∞∑j=1lN‾0(r,1f1−αj)∑j=1lN‾0(r,1f2−αj)>52l−5, then $f_{1}\equiv f_{2}$f1≡f2. Our results are some improvements and extension of previous theorems given by Cao–Yi and Ge–Wu.

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