scholarly journals A question of Gross and the uniqueness of entire functions

1995 ◽  
Vol 138 ◽  
pp. 169-177 ◽  
Author(s):  
Hong-Xun yi
Keyword(s):  
Entire Function ◽  
Nevanlinna Theory ◽  
Entire Functions ◽  
Linear Measure ◽  
Finite Linear

For any set S and any entire function f letwhere each zero of f — a with multiplicity m is repeated m times in Ef(S) (cf. [1]). It is assumed that the reader is familiar with the notations of the Nevanlinna Theory (see, for example, [2]). It will be convenient to let E denote any set of finite linear measure on 0 < r < ∞, not necessarily the same at each occurrence. We denote by S(r, f) any quantity satisfying .

scholarly journals A uniqueness problem for entire functions related to Brück’s conjecture

2018 ◽  
Vol 68 (4) ◽  
pp. 823-836
Author(s):  
Nguyen Van Thin ◽  
Ha Tran Phuong ◽  
Leuanglith Vilaisavanh
Keyword(s):  
Entire Function ◽  
Uniqueness Theorem ◽  
Nevanlinna Theory ◽  
Meromorphic Functions ◽  
Normal Family ◽  
Entire Functions ◽  
Uniqueness Problem ◽  
Family Theory

Abstract In this paper, we prove a normal criteria for family of meromorphic functions. As an application of that result, we establish a uniqueness theorem for entire function concerning a conjecture of R. Brück. The above uniqueness theorem is an improvement of a problem studied by L. Z. Yang et al. [14]. However, our method differs the method of L. Z. Yang et al. [14]. We mainly use normal family theory and combine it with Nevanlinna theory instead of using only the Nevanlinna theory as in [14].

scholarly journals Uniqueness on entire functions and their nth order exact differences with two shared values

2020 ◽  
Vol 18 (1) ◽  
pp. 211-215
Author(s):  
Shengjiang Chen ◽  
Aizhu Xu
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Shared Values ◽  
Simple Method ◽  
Exact Difference ◽  
Hyper Order

Abstract Let f(z) be an entire function of hyper order strictly less than 1. We prove that if f(z) and its nth exact difference {\Delta }_{c}^{n}f(z) share 0 CM and 1 IM, then {\Delta }_{c}^{n}f(z)\equiv f(z) . Our result improves the related results of Zhang and Liao [Sci. China A, 2014] and Gao et al. [Anal. Math., 2019] by using a simple method.

scholarly journals Primeable entire functions

1973 ◽  
Vol 51 ◽  
pp. 123-130 ◽  
Author(s):  
Fred Gross ◽  
Chung-Chun Yang ◽  
Charles Osgood
Keyword(s):  
Entire Function ◽  
Periodic Function ◽  
Rational Function ◽  
Entire Functions ◽  
Left And Right

An entire function F(z) = f(g(z)) is said to have f(z) and g(z) as left and right factors respe2tively, provided that f(z) is meromorphic and g(z) is entire (g may be meromorphic when f is rational). F(z) is said to be prime (pseudo-prime) if every factorization of the above form implies that one of the functions f and g is bilinear (a rational function). F is said to be E-prime (E-pseudo prime) if every factorization of the above form into entire factors implies that one of the functions f and g is linear (a polynomial). We recall here that an entire non-periodic function f is prime if and only if it is E-prime [5]. This fact will be useful in the sequel.

The polynomial hull of a set of finite linear measure in Cn

1986 ◽  
Vol 47 (1) ◽  
pp. 238-242 ◽  
Author(s):  
H. Alexander
Keyword(s):  
Linear Measure ◽  
Polynomial Hull ◽  
Finite Linear

The distribution of the values of an entire function whose coefficients are independent random variables (II)

1995 ◽  
Vol 118 (3) ◽  
pp. 527-542 ◽  
Author(s):  
A. C. Offord
Keyword(s):  
Entire Function ◽  
Uniform Distribution ◽  
Entire Functions ◽  
Random Variables ◽  
Independent Random Variables ◽  
Almost All

SummaryThis is a study of entire functions whose coefficients are independent random variables. When the space of such functions is symmetric it is shown that independence of the coefficients alone is sufficient to ensure that almost all such functions will, for large z, be large except in certain small neighbourhoods of the zeros called pits. In each pit the function takes every not too large value and these pits have a certain uniform distribution.

scholarly journals Linear measure on plane continua of finite linear measure

1989 ◽  
Vol 27 (1-2) ◽  
pp. 169-177
Author(s):  
H. Alexander
Keyword(s):  
Linear Measure ◽  
Finite Linear

scholarly journals NONDECREASING FUNCTIONS, EXCEPTIONAL SETS AND GENERALIZED BOREL LEMMAS

2010 ◽  
Vol 88 (3) ◽  
pp. 353-361
Author(s):  
R. G. HALBURD ◽  
R. J. KORHONEN
Keyword(s):  
Maximum Modulus ◽  
Point Of View ◽  
Modulus Function ◽  
Linear Measure ◽  
Real Analysis ◽  
Exceptional Set ◽  
Exceptional Sets ◽  
Finite Linear ◽  
Increasing Function ◽  
Nondecreasing Functions

AbstractAccording to the classical Borel lemma, any positive nondecreasing continuous function T satisfiesT(r+1/T(r))≤2T(r) outside a possible exceptional set of finite linear measure. This lemma plays an important role in the theory of entire and meromorphic functions, where the increasing function T is either the logarithm of the maximum modulus function, or the Nevanlinna characteristic. As a result, exceptional sets appear throughout Nevanlinna theory, in particular in Nevanlinna’s second main theorem. In this paper, we consider generalizations of Borel’s lemma. Conversely, we consider ways in which certain inequalities can be modified so as to remove exceptional sets. All results discussed are presented from the point of view of real analysis.

scholarly journals A Class of functional equations which have entire solutions

1988 ◽  
Vol 38 (3) ◽  
pp. 351-356 ◽  
Author(s):  
Peter L. Walker
Keyword(s):  
Functional Equation ◽  
Entire Function ◽  
Wide Class ◽  
Functional Equations ◽  
Entire Functions ◽  
Inverse Function ◽  
Entire Solution ◽  
Entire Solutions ◽  
Image Position

We consider the Abelian functional equationwhere φ is a given entire function and g is to be found. The inverse function f = g−1 (if one exists) must satisfyWe show that for a wide class of entire functions, which includes φ(z) = ez − 1, the latter equation has a non-constant entire solution.

scholarly journals A Remark on Small Values of Entire Functions

1966 ◽  
Vol 15 (2) ◽  
pp. 121-123 ◽  
Author(s):  
S. L. Segal
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Image Position

Let f(z) be an entire function, M(r) the maximum of f(z) on ∣z∣=r, and λ>1. Let Eλ=Eλ(f{z:log∣f(z)≦(1-λ)log(M∣z∣)}, and denote the density of Eλbywhere m is planar measure.

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