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.

Derivatives of Meromorphic Functions with Multiple Zeros and Rational Functions

2007 ◽  
Vol 8 (2) ◽  
pp. 483-491 ◽  
Author(s):  
Xuecheng Pang ◽  
Shahar Nevo ◽  
Lawrence Zalcman
Keyword(s):  
Meromorphic Functions ◽  
Rational Functions ◽  
Multiple Zeros ◽  
Derivatives Of

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 .

The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions II

2000 ◽  
Vol 20 (3) ◽  
pp. 895-910 ◽  
Author(s):  
GWYNETH M. STALLARD
Keyword(s):  
Continuous Function ◽  
Hausdorff Dimension ◽  
Meromorphic Function ◽  
Statistical Mechanics ◽  
Rational Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Real Analytic ◽  
Analytic Maps

Ruelle (Repellers for real analytic maps. Ergod. Th. & Dynam. Sys.2 (1982), 99–108) used results from statistical mechanics to show that, when a rational function $f$ is hyperbolic, the Hausdorff dimension of the Julia set, $\dim J(f)$, depends real analytically on $f$. We give a proof of the fact that $\dim J(f)$ is a continuous function of $f$ that does not depend on results from statistical mechanics and we show that this result can be extended to a class of transcendental meromorphic functions. This enables us to show that, for each $d \in (0,1)$, there exists a transcendental meromorphic function $f$ with $\dim J(f) = d$.

Non-real zeros of derivatives of real meromorphic functions of infinite order

2010 ◽  
Vol 150 (2) ◽  
pp. 343-351 ◽  
Author(s):  
J. K. LANGLEY
Keyword(s):  
Meromorphic Function ◽  
Infinite Order ◽  
Meromorphic Functions ◽  
Real Zeros ◽  
Derivatives Of ◽  
Real Poles ◽  
Zeros Of Derivatives

AbstractLet f be a real meromorphic function of infinite order in the plane, with finitely many zeros and non-real poles. Then f″ has infinitely many non-real 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.

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).

scholarly journals Value Distribution of Meromorphic Solutions and Their Derivatives of Complex Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Abdallah El Farissi
Keyword(s):  
Differential Equations ◽  
Meromorphic Functions ◽  
Higher Order ◽  
Value Distribution ◽  
Meromorphic Solutions ◽  
Small Functions ◽  
Complex Differential Equations ◽  
Linear Differential ◽  
The Relationship ◽  
Derivatives Of

We deal with the relationship between the small functions and the derivatives of solutions of higher-order linear differential equations f(k)+Ak-1f(k-1)+⋯+A0f=0,   k≥2, where Aj(z)  (j=0,1,…,k-1) are meromorphic functions. The theorems of this paper improve the previous results given by El Farissi, Belaïdi, Wang, Lu, Liu, and Zhang.

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