scholarly journals Logarithmic derivative estimates of meromorphic functions of finite order in the half-plane

2020 ◽  
Vol 54 (2) ◽  
pp. 172-187
Author(s):  
I.E. Chyzhykov ◽  
A.Z. Mokhon'ko
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Half Plane ◽  
Meromorphic Functions ◽  
Logarithmic Derivative ◽  
Sharp Estimates ◽  
Exceptional Sets ◽  
Upper Half Plane ◽  
Logarithmic Derivatives

We established new sharp estimates outside exceptional sets for of the logarithmic derivatives $\frac{d^ {k} \log f(z)}{dz^k}$ and its generalization $\frac{f^{(k)}(z)}{f^{(j)}(z)}$, where $f$ is a meromorphic function $f$ in the upper half-plane, $k>j\ge0$ are integers. These estimates improve known estimates due to the second author in the class of meromorphic functions of finite order.Examples show that size of exceptional sets are best possible in some sense.

scholarly journals On Tangential Principal Cluster Sets of Normal Meromorphic Functions

1970 ◽  
Vol 40 ◽  
pp. 133-137 ◽  
Author(s):  
Theodore A. Vessey
Keyword(s):  
Meromorphic Function ◽  
Conformal Mapping ◽  
Finite Order ◽  
Real Axis ◽  
Half Plane ◽  
Meromorphic Functions ◽  
The Family ◽  
Normal Meromorphic Function ◽  
Upper Half Plane ◽  
Principal Cluster

Let w = f(z) be a normal meromorphic function defined in the upper half plane U = {Im(z) >0}. We recall that a meromorphic function f(z) is normal in U if the family {f(S(z))}, where z′ = S(z) is an arbitrary one-one conformal mapping of U onto U, is normal in the sense of Montel. It is the purpose of this paper to state some results on the behavior of f(z) on curves which approach a point x0 on the real axis R with a fixed (finite) order of contact q at x0.

scholarly journals Extreme problems in the space of meromorphic functions of finite order in the half plane. II

2020 ◽  
Vol 54 (2) ◽  
pp. 154-161
Author(s):  
K.G. Malyutin ◽  
A.A. Revenko
Keyword(s):  
Meromorphic Function ◽  
Half Plane ◽  
Fourier Coefficients ◽  
Meromorphic Functions ◽  
Sharp Estimate ◽  
Extremal Problems ◽  
Parseval Equality ◽  
Upper Limits ◽  
Upper Half Plane ◽  
Space Of Meromorphic Functions

The extremal problems in the space of meromorphic functions of order $\rho>0$ in upper half-plane are studed.The method for studying is based on the theory of Fourier coefficients of meromorphic functions. The concept of just meromorphic function of order $\rho>0$ in upper half-plane is introduced. Using Lemma on the P\'olya peaks and the Parseval equality, sharp estimate from below of the upper limits of relations Nevanlinna characteristics of meromorphic functions in the upper half plane are obtained.

scholarly journals The meromorphic functions of completely regular growth on the upper half-plane

2020 ◽  
Vol 30 (3) ◽  
pp. 396-409
Author(s):  
K.G. Malyutin ◽  
M.V. Kabanko
Keyword(s):  
Meromorphic Function ◽  
Half Plane ◽  
Meromorphic Functions ◽  
Growth Function ◽  
Regular Growth ◽  
Completely Regular ◽  
Upper Half Plane ◽  
Definition Of ◽  
Half Axis ◽  
Increasing Function

A strictly positive continuous unbounded increasing function $\gamma(r)$ on the half-axis $[0,+\infty)$ is called growth function. Let the growth function $\gamma(r)$ satisfies the condition $\gamma(2r)\leq M\gamma(r)$ for some $M>0$ and for all $r>0$. In the paper, the class $JM(\gamma(r))^o$ of meromorphic functions of completely regular growth on the upper half-plane with respect to the growth function $\gamma$ is considered. The criterion for the meromorphic function $f$ to belong to the space $JM(\gamma(r))^o$ is obtained. The definition of the indicator of function from the space $JM(\gamma(r))^o$ is introduced. It is proved that the indicator belongs to the space $\mathbf{L}^p[0,\pi]$ for all $p>1$.

scholarly journals On Properties of Differences Polynomials about Meromorphic Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Jianming Qi ◽  
Jie Ding ◽  
Wenjun Yuan
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Special Class ◽  
Meromorphic Functions ◽  
Value Distribution ◽  
Class Difference ◽  
Difference Polynomial

We study the value distribution of a special class difference polynomial about finite order meromorphic function. Our methods of the proof are also different from ones in the previous results by Chen (2011), Liu and Laine (2010), and Liu and Yang (2009).

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 .

scholarly journals On Complex Explicit Formulae Connected with the Möbius Function of an Elliptic Curve

2014 ◽  
Vol 57 (2) ◽  
pp. 381-389
Author(s):  
Adrian Łydka
Keyword(s):  
Functional Equation ◽  
Meromorphic Function ◽  
Elliptic Curve ◽  
Explicit Formula ◽  
Half Plane ◽  
Complex Plane ◽  
Möbius Function ◽  
Mobius Function ◽  
Upper Half Plane ◽  
Explicit Formulae

AbstractWe study analytic properties function m(z, E), which is defined on the upper half-plane as an integral from the shifted L-function of an elliptic curve. We show that m(z, E) analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for m(z, E) in the strip |ℑz| < 2π.

On Uniqueness of Dirichlet Series

2020 ◽  
Author(s):  
Bao Qin Li
Keyword(s):  
Analytic Continuation ◽  
Dirichlet Series ◽  
Finite Order ◽  
Half Plane ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Uniqueness Problem ◽  
Uniqueness Theorems

Abstract We give a characterization of the ratio of two Dirichelt series convergent in a right half-plane under an analytic condition, which is applicable to a uniqueness problem for Dirichlet series admitting analytic continuation in the complex plane as meromorphic functions of finite order; uniqueness theorems are given in terms of a-points of the functions.

scholarly journals Value distribution and the Lemma of the logarithmic derivative on polydiscs

1983 ◽  
Vol 6 (4) ◽  
pp. 617-669 ◽  
Author(s):  
Wilhelm Stoll
Keyword(s):  
Distribution Function ◽  
Meromorphic Functions ◽  
Logarithmic Derivative ◽  
Value Distribution ◽  
Exceptional Sets ◽  
Vector Variable

Value distribution is developed on polydiscs with the special emphasis that the value distribution function depend on a vector variable. A Lemma of the logarithmic derivative for meromorphic functions on polydiscs is derived. Here the Bergman boundary of the polydiscs is approached along cones of any dimension and exceptional sets for such an approach are defined.

scholarly journals Filling Disks of Hayman Type of Meromorphic Functions

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Nan Wu ◽  
Zuxing Xuan
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Meromorphic Functions

We obtain the existence of the filling disks with respect to Hayman directions. We prove that, under the conditionlimsupr→∞⁡(Tr,f/log⁡r3)=∞, there exists a sequence of filling disks of Hayman type, and these filling disks can determine a Hayman direction. Every meromorphic function of positive and finite orderρhas a sequence of filling disks of Hayman type, which can also determine a Hayman direction of orderρ.

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).

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