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.

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.

On the zeros and Fourier transforms of entire functions in the Paley–Wiener space

1996 ◽  
Vol 119 (2) ◽  
pp. 357-362 ◽  
Author(s):  
Konstantin M. Dyakonov
Keyword(s):  
Entire Function ◽  
Complex Number ◽  
Compact Support ◽  
Fourier Transforms ◽  
Entire Functions ◽  
Wiener Space ◽  
Dirichlet Integral ◽  
Fixed Complex ◽  
Image Position

AbstractLet f be an entire function of the formwhere ø is a function in L2(ℝ) with compact support. If f|ℝ is real-valued then, for obvious reasons, (a) the supporting interval for ø is symmetric with respect to the origin, andAssuming that f has no zeros in {Im z > 0}, we prove that the converse is also true: (a) and (b) together imply that f|ℝ takes values in αℝ, where α is a fixed complex number.The proof relies on a certain formula involving the Dirichlet integral, which may be interesting on its own.

Representation Formulas for Integrable and Entire Functions of Exponential Type II

1991 ◽  
Vol 43 (1) ◽  
pp. 34-47 ◽  
Author(s):  
Clément Frappier
Keyword(s):  
Entire Function ◽  
Real Axis ◽  
Exponential Type ◽  
Entire Functions ◽  
Type Ii ◽  
The Real ◽  
Representation Formulas ◽  
Functions Of Exponential Type ◽  
Image Position

We adopt the terminology and notations of [5]. If f ∈ Bτ is an entire function of exponential type τ bounded on the real axis then we have the complementary interpolation formulas [1, p. 142-143] andwhere t, γ are reals and

The Second Mean Values of Entire Functions

1966 ◽  
Vol 18 ◽  
pp. 1113-1120
Author(s):  
Q. I. Rahman
Keyword(s):  
Entire Function ◽  
Convex Function ◽  
Dirichlet Series ◽  
Complex Variable ◽  
Entire Functions ◽  
Mean Values ◽  
Image Position

Let f(z) be an entire function of the complex variable z = x + iy defined by the everywhere absolutely convergent Dirichlet series1.1Ifthen log m(x,f) is an increasing convex function of x (2), andis called the Ritt order of f(z).

Inequalities for Entire Functions of Exponential Type

1984 ◽  
Vol 27 (4) ◽  
pp. 463-471 ◽  
Author(s):  
Clément Frappier
Keyword(s):  
Entire Function ◽  
Real Axis ◽  
Exponential Type ◽  
Entire Functions ◽  
Indicator Function ◽  
Trigonometric Polynomials ◽  
Bernstein’S Inequality ◽  
The Real ◽  
Functions Of Exponential Type ◽  
Image Position

AbstractBernstein's inequality says that if f is an entire function of exponential type τ which is bounded on the real axis thenGenchev has proved that if, in addition, hf (π/2) ≤0, where hf is the indicator function of f, thenUsing a method of approximation due to Lewitan, in a form given by Hörmander, we obtain, to begin, a generalization and a refinement of Genchev's result. Also, we extend to entire functions of exponential type two results first proved for polynomials by Rahman. Finally, we generalize a theorem of Boas concerning trigonometric polynomials vanishing at the origin.

scholarly journals Some representation formulae for entire functions of exponential type

1988 ◽  
Vol 37 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Clément Frappier
Keyword(s):  
Entire Function ◽  
Exponential Type ◽  
Entire Functions ◽  
Bernoulli Numbers ◽  
First Case ◽  
Functions Of Exponential Type ◽  
Explicit Formulae ◽  
Image Position ◽  
Fejér Mean ◽  
Derivatives Of

We obtain some explicit formulae for series of the typewhere f is an entire function of exponential type τ, bounded on the real exis (and satisfying in the first case). These series are expressed in terms of the derivatives of f and Bernoulli numbers. We examine the case where f is a trigonometric polynomial which lead us, in particular, to a new representation of the associated Fejér mean.

Approximation and Interpolation by Entire Functions of Several Variables

2010 ◽  
Vol 53 (1) ◽  
pp. 11-22 ◽  
Author(s):  
Maxim R. Burke
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Nondecreasing Sequence ◽  
Real Numbers ◽  
Several Variables ◽  
Accumulation Points ◽  
Image Position

AbstractLet f : ℝn → ℝ be C∞ and let h: ℝn → ℝ be positive and continuous. For any unbounded nondecreasing sequence ﹛ck﹜ of nonnegative real numbers and for any sequence without accumulation points ﹛xm﹜ in ℝn, there exists an entire function g : ℂn → ℂ taking real values on ℝn such thatThis is a version for functions of several variables of the case n = 1 due to L. Hoischen.

scholarly journals Approximation of entire functions over Carathéodory domains

1982 ◽  
Vol 25 (2) ◽  
pp. 221-229 ◽  
Author(s):  
G.P. Kapoor ◽  
A. Nautiyal
Keyword(s):  
Entire Function ◽  
Analytic Continuation ◽  
Finite Type ◽  
Finite Order ◽  
Jordan Curve ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Approximation Error ◽  
Image Position ◽  
Necessary And Sufficient

Let D be a domain bounded by a Jordan curve. For 1 ≤ p ≤ ∞, let Lp(D) be the class of all functions f holomorphic in D such that where A is the area of D. For f ∈Lp(D), setπn consists of all polynomials of degree at most n. Recently, Andre Giroux (J. Approx. Theory 28 (1980), 45–53) has obtained necessary and sufficient conditions, in terms of the rate of decrease of the approximation error , such that has an analytic continuation as an entire function having finite order and finite type. In the present paper we have considered the approximation error (*) on a Carathéodory domain and have extended the results of Giroux for the case 1 ≤ p < 2.

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.

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