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 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.

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.

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 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).

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.

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.

scholarly journals Fixed points of differences of meromorphic functions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zhaojun Wu ◽  
Jia Wu
Keyword(s):  
Meromorphic Function ◽  
Fixed Points ◽  
Finite Order ◽  
Complex Number ◽  
Meromorphic Functions ◽  
Acta Math ◽  
Transcendental Meromorphic Function ◽  
Nonzero Complex Number ◽  
Discrete Analogues

Abstract Let f be a transcendental meromorphic function of finite order and c be a nonzero complex number. Define $\Delta _{c}f=f(z+c)-f(z)$ Δ c f = f ( z + c ) − f ( z ) . The authors investigate the existence on the fixed points of $\Delta _{c}f$ Δ c f . The results obtained in this paper may be viewed as discrete analogues on the existing theorem on the fixed points of $f'$ f ′ . The existing theorem on the fixed points of $\Delta _{c}f$ Δ c f generalizes the relevant results obtained by (Chen in Ann. Pol. Math. 109(2):153–163, 2013; Zhang and Chen in Acta Math. Sin. New Ser. 32(10):1189–1202, 2016; Cui and Yang in Acta Math. Sci. 33B(3):773–780, 2013) et al.

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 ZEROS OF DIFFERENTIAL POLYNOMIALS IN REAL MEROMORPHIC FUNCTIONS

2005 ◽  
Vol 48 (2) ◽  
pp. 279-293 ◽  
Author(s):  
Walter Bergweiler ◽  
Alex Eremenko ◽  
Jim K. Langley
Keyword(s):  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Rational Functions ◽  
Differential Polynomials ◽  
Real Zeros ◽  
Transcendental Meromorphic Function

AbstractWe investigate whether differential polynomials in real transcendental meromorphic functions have non-real zeros. For example, we show that if $g$ is a real transcendental meromorphic function, $c\in\mathbb{R}\setminus\{0\}$ and $n\geq3$ is an integer, then $g'g^n-c$ has infinitely many non-real zeros. If $g$ has only finitely many poles, then this holds for $n\geq2$. Related results for rational functions $g$ are also considered.

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