Borel Directions Research Articles

On Common Borel Direction of Entire Function f and Its q-Difference Operator

The common Borel direction of entire function f(z) and its q-difference operator is studied by using Nevanlinna theory. Some conditions of the existence of common Borel direction for entire function f(z) and its q-difference operator are obtained in this paper.

ON THE ITERATIONS AND THE ARGUMENT DISTRIBUTION OF MEROMORPHIC FUNCTIONS

AbstractThis paper consists of two parts. The first is to study the existence of a point a at the intersection of the Julia set and the escaping set such that a goes to infinity under iterates along Julia directions or Borel directions. Additionally, we find such points that approximate all Borel directions to escape if the meromorphic functions have positive lower order. We confirm the existence of such slowly escaping points under a weaker growth condition. The second is to study the connection between the Fatou set and argument distribution. In view of the filling disks, we show nonexistence of multiply connected Fatou components if an entire function satisfies a weaker growth condition. We prove that the absence of singular directions implies the nonexistence of large annuli in the Fatou set.

General Relationship between the Total Number of Deficient Values of Meromorphic Functions and the Total Number of Borel Directions Guides Evolutionary Calculation Research

General Relationship between the Total Number of Deficient Values of Meromorphic Functions and the Total Number of Borel Directions Guides Evolutionary Calculation Research

Meromorphic functions sharing three values with their derivatives in an angular domains

In this paper, we investigate the uniqueness of transcendental meromorphic functions sharing three values with their derivatives in an arbitrary small angular domain including a Borel direction. The obtained results extend the corresponding results from Gundersenand and Mues-Steinmetz, Zheng and Li-Liu-Yi, Chen.

Borel directions and the uniqueness of algebroid functions

In this paper, using Nevanlinna theory we discuss the relations between the Borel directions and the uniqueness of algebroid functions. We get several uniqueness theorems of algebroid functions in angular domain which contains the Borel directions. These results extend those for meromorphic functions obtained by some scholars.

Borel Directions and the Uniqueness of Algebroid Functions Dealing with Multiple Values

In this paper, we investigate the uniqueness of algebroid functions in angular domain by the method of conformal mapping. We discuss the relations between the Borel directions and uniquenss with the multiple values of algebroid functions and obtain some results which extend some uniqueness results of meromorphic functions to that of algebroid functions.

Generalization of P\xf3lya\u2019s Zero Distribution Theory for Exponential Polynomials, and Sharp Results for Asymptotic Growth

An exponential polynomial of order q is an entire function of the form f(z)=P1(z)eQ1(z)+⋯+Pk(z)eQk(z),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} f(z)=P_1(z)e^{Q_1(z)}+\\cdots +P_k(z)e^{Q_k(z)}, \\end{aligned}$$\\end{document}where the coefficients P_j(z),Q_j(z) are polynomials in z such that max{deg(Qj)}=q.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\max \\{\\deg (Q_j)\\}=q. \\end{aligned}$$\\end{document}In 1977 Steinmetz proved that the zeros of f lying outside of finitely many logarithmic strips around so called critical rays have exponent of convergence le q-1. This result does not say anything about the zero distribution of f in each individual logarithmic strip. Here, it is shown that the asymptotic growth of the non-integrated counting function of zeros of f is asymptotically comparable to r^q in each logarithmic strip. The result generalizes the first order results by Pólya and Schwengeler from the 1920’s, and it shows, among other things, that the critical rays of f are precisely the Borel directions of order q of f. The error terms in the asymptotic equations for T(r, f) and N(r, 1/f) originally due to Steinmetz are also improved.

Distributions of Branch Points of Some Algebroid Functions

In this paper, we investigate the distribution of branch points of some algebroid functions with infinitely many branch points. A sufficient condition that algebroid function has infinitely many branch points is given. By the Ahlfors' theory of covering surfaces, we also obtain the relationship between the Borel cluster directions of branch points and the Borel directions.

Borel directions of solutions of a second order linear complex differential equation

Borel directions of solutions of a second order linear complex differential equation

The radial oscillation of entire solutions of complex differential equations

This paper is devoted to studying the radial oscillation of solutions of linear differential equations f″+A(z)f′+B(z)f=F(z), where A(z), B(z) and F(z) are transcendental entire functions. With some added conditions on coefficients, we show that there exists some close relation between the Borel directions of solutions and that of F(z), and also obtain some estimates of growth of solutions in the angular domains.

The Borel directions of algebroidal function and its coefficient functions

In this article, the relationship between the Borel direction of algebroidal function and its coefficient functions is studied for the first time. To begin with, several theorems of algebroidal functions in unit disk are proved. By these theorems, some interesting conclusions are obtained.

Some inequalities and applications on Borel direction and exceptional values of meromorphic functions

In view of the Nevanlinna theory in the angular domain, we study the exceptional values of meromorphic functions in the Borel direction and also establish some inequalities on the exceptional values of meromorphic functions in the Borel direction. Based on these inequalities, we also give two theorems and some corollaries as regards exceptional values of meromorphic functions in the Borel direction.MSC:30D30, 30D35.

Some notes on the Borel directions of meromorphic functions

We prove that the distribution of the Borel radii (indirect Borel points) and that of Borel radii (indirect Borel points) concerning the small functions of a meromorphic function are the same. Furthermore, some equivalent conclusions on the Borel radii (indirect Borel points) of meromorphic functions of order $0<\rho<\infty$ are established. This is a continuous work of Tsuji [4,5].

On the growth of solutions of a class of second-order complex differential equations

In this paper, we consider the differential equation , where and are meromorphic functions, is a non-constant polynomial. Assume that has an infinite deficient value and finitely many Borel directions. We give some conditions on which guarantee that every solution of the equation has infinite order. MSC:34AD20, 30D35.

Borel's directions and shared values

In this article, we study the problems of Borel's directions of meromorphic functions concerning shared values and prove that if two meromorphic functions of infinite order share three distinct values, their Borel's directions are same.

Borel Directions and Uniqueness of Meromorphic Functions

We investigate the relationship between Borel directions and uniqueness of meromorphic functions and obtain some results of meromorphic functions sharing four distinct values IM and one set in an angular domain containing a Borel line. Our result is an improvement of a recent theorem given by Long and Wu (2012).

Meromorphic functions of infinite order in the angular domain

Abstract We investigate the value distribution of meromorphic functions in the angular domain. In particular, we show a close relationship between radially distributed values and Borel directions of transcendental meromorphic functions of infinite order in some angular domains.

On value distribution of meromorphic functions with respect to arguments I

The theory of value distribution of meromorphic functions is developed in two directions: module distribution and argument distribution. This article is a survey on two aspects of the value distribution of meromorphic functions with respect to arguments and it also includes some new results. The first part is to survey a characterization of growth of a transcendental meromorphic function for which the number of a-points in certain curvilinear sectors, specially in angular domains, is restricted. The second part is devoted to the discussion of singular directions: the Julia, Borel directions and T-directions. In the final section, some problems will be given for further discussion.

ON FILLING DISCS IN THE STRONG BOREL DIRECTION OF ALGEBROID FUNCTION WITH FINITE ORDER

Using Ahlfors' covering surface method, some properties on the strong Borel direction of algebroid function of finite order are obtained. The main object of this paper is to prove existence theorem of a strong Borel direction and the existence of filling discs in such direction which briefly extends some results of meromorphic function.

THE ZEROS DISTRIBUTION OF SOLUTIONS OF HIGHER ORDER DIFFERENTIAL EQUATIONS IN AN ANGULAR DOMAIN

In this paper, we investigate the zeros distribution and Borel direction for the solutions of linear homogeneous differential equation <TEX>$f^{(n)}+A_{n-2}(z)f^{(n-2)}+{\cdots}+A_1(z)f'+A_0(z)f=0(n{\geq}2)$</TEX> in an angular domain. Especially, we establish a relation between a cluster ray of zeros and Borel direction.