On a generalized Eulerian product defining a meromorphic function on the whole complex plane

2019 ◽  
Vol 20 (2) ◽  
pp. 148-160
Author(s):  
Nikolai Nikolaevich Dobrovol'skii ◽  
Mikhail Nikolaevich Dobrovol'skii ◽  
Nikolai Mihailovich Dobrovol'skii
Keyword(s):  
Meromorphic Function ◽  
Complex Plane

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.

scholarly journals Milloux Inequality ofE-Valued Meromorphic Function

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Zhaojun Wu ◽  
Zuxing Xuan
Keyword(s):  
Banach Space ◽  
Meromorphic Function ◽  
Complex Plane ◽  
Complex Banach Space ◽  
Schauder Basis ◽  
Infinite Dimensional ◽  
Borel Exceptional Values

The main purpose of this paper is to establish the Milloux inequality ofE-valued meromorphic function from the complex planeℂto an infinite dimensional complex Banach spaceEwith a Schauder basis. As an application, we study the Borel exceptional values of anE-valued meromorphic function and those of its derivatives; results are obtained to extend some related results for meromorphic scalar-valued function of Singh, Gopalakrishna, and Bhoosnurmath.

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 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?

scholarly journals A Property of the Principal Cluster sets of a Class of Holomorphic Functions

1971 ◽  
Vol 43 ◽  
pp. 157-159
Author(s):  
F. Bagemihl
Keyword(s):  
Meromorphic Function ◽  
Unit Circle ◽  
Unit Disk ◽  
Complex Plane ◽  
Riemann Sphere ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Open Unit Disk ◽  
Cluster Set ◽  
Principal Cluster

Let D be the open unit disk and Γ be the unit circle in the complex plane, and denote by Ω the Riemann sphere. If f(z) is a meromorphic function in D, and if ζ∈Г, then the principal cluster set of f at ζ is the set

A note on Hayman’s conjecture

2020 ◽  
Vol 31 (06) ◽  
pp. 2050048
Author(s):  
Ta Thi Hoai An ◽  
Nguyen Viet Phuong
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Complex Plane ◽  
Function Theory ◽  
Differential Polynomials ◽  
Research Problems

In this paper, we will give suitable conditions on differential polynomials [Formula: see text] such that they take every finite nonzero value infinitely often, where [Formula: see text] is a meromorphic function in complex plane. These results are related to Problems 1.19 and 1.20 in a book of Hayman and Lingham [Research Problems in Function Theory, preprint (2018), https://arxiv.org/pdf/1809.07200.pdf ]. As consequences, we give a new proof of the Hayman conjecture. Moreover, our results allow differential polynomials [Formula: see text] to have some terms of any degree of [Formula: see text] and also the hypothesis [Formula: see text] in [Theorem 2 of W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana 11(2) (1995) 355–373] is replaced by [Formula: see text] in our result.

On Isomorphisms of Meromorphic Function Fields

1968 ◽  
Vol 20 ◽  
pp. 1230-1241 ◽  
Author(s):  
James J. Kelleher
Keyword(s):  
Meromorphic Function ◽  
Complex Plane ◽  
Algebraic Structure ◽  
Riemann Surfaces ◽  
Function Fields ◽  
Plane Region ◽  
Algebraic Properties ◽  
Compact Riemann Surfaces ◽  
Conformal Equivalence ◽  
Ring Of Functions

In this work we deal with algebraic properties of some fields of functions meromorphic in the complex plane with a view to determining the possible isomorphisms between two such fields. Interest in problems of this type began with a paper by Bers (2), in which it was shown that the algebraic structure of the ring of functions analytic on a plane region determines the conformai structure of the region to within conformai or anti-conformal equivalence, and this result was later extended to arbitrary non-compact Riemann surfaces by Nakai (7).

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.

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