On the Pólya conjecture concerning the maximum and minimum of the modulus of an entire function of finite order given by a lacunary power series

1990 ◽  
Vol 16 (2) ◽  
pp. 143-157 ◽  
Author(s):  
O. B. Skaskiv
Keyword(s):  
Entire Function ◽  
Power Series ◽  
Finite Order

On regularly varying moments for discrete laws

2010 ◽  
Vol 47 (1) ◽  
pp. 118-126
Author(s):  
Slavko Simic
Keyword(s):  
Entire Function ◽  
Power Series ◽  
Finite Order ◽  
Asymptotic Relation ◽  
Slowly Varying Function ◽  
Regularly Varying ◽  
Power Series Distributions

For a discrete law F and large n , we investigate the asymptotic relation \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$EX_n^\mu \ell (X_n ) \sim C_\mu (EX_n )^\mu \in (a,b),C_\mu > 0(EX_n \to \infty ),$$ \end{document}, where ℓ(·) is an arbitrary slowly varying function. By a result from [6], it follows that regularly varying moments for Power Series Distributions, generated by an entire function of finite order, satisfy the above relation with Cµ = 1 for each µ ∈ (0, ∞).

scholarly journals On regularly varying moments for power series distributions

2006 ◽  
Vol 80 (94) ◽  
pp. 253-258 ◽  
Author(s):  
Slavko Simic
Keyword(s):  
Asymptotic Behavior ◽  
Entire Function ◽  
Power Series ◽  
Finite Order ◽  
Slowly Varying Function ◽  
Power Series Distribution ◽  
Regularly Varying ◽  
Power Series Distributions

For the power series distribution, generated by an entire function of finite order, we obtain the asymptotic behavior of its regularly varying moments. Namely, we prove that EwX??(X)\sim(EwX)??(EwX), ? > 0 (w??), where ?(?) is an arbitrary slowly varying function.

A NOTE ON LACUNARY POWER SERIES WITH RATIONAL COEFFICIENTS

2015 ◽  
Vol 93 (3) ◽  
pp. 372-374 ◽  
Author(s):  
DIEGO MARQUES ◽  
JOSIMAR RAMIREZ ◽  
ELAINE SILVA
Keyword(s):  
Entire Function ◽  
Power Series

In this note, we prove that for any ${\it\nu}>0$, there is no lacunary entire function $f(z)\in \mathbb{Q}[[z]]$ such that $f(\mathbb{Q})\subseteq \mathbb{Q}$ and $\text{den}f(p/q)\ll q^{{\it\nu}}$, for all sufficiently large $q$.

scholarly journals The Spaces of Entire Function of Finite Order

2016 ◽  
Vol 7 (3) ◽  
Author(s):  
Malyutin KG ◽  
Studenikina IG
Keyword(s):  
Entire Function ◽  
Finite Order

On the question of whether f″+ e−zf′ + B(z)f = 0 can admit a solution f ≢ 0 of finite order

1986 ◽  
Vol 102 (1-2) ◽  
pp. 9-17 ◽  
Author(s):  
Gary G. Gundersen
Keyword(s):  
Entire Function ◽  
Finite Order ◽  
Infinite Order ◽  
Independent Interest ◽  
Partial Result ◽  
Transcendental Entire Function ◽  
Degree Polynomial ◽  
Odd Degree ◽  
Polynomial Of Odd Degree

SynopsisWe show that if B(z) is either (i) a transcendental entire function with order (B)≠1, or (ii) a polynomial of odd degree, then every solution f≠0 to the equation f″ + e−zf′ + B(z)f = 0 has infinite order. We obtain a partial result in the case when B(z) is an even degree polynomial. Our method of proof and lemmas for case (i) of the above result have independent interest.

scholarly journals Bank-Laine Functions with Real Zeros

2020 ◽  
Vol 20 (3-4) ◽  
pp. 653-665
Author(s):  
J. K. Langley
Keyword(s):  
Entire Function ◽  
Finite Order ◽  
Explicit Representation ◽  
Trigonometric Functions ◽  
Exponent Of Convergence ◽  
Real Zeros ◽  
Real Entire Function ◽  
Bounded Below ◽  
Quasiconformal Surgery

AbstractSuppose that E is a real entire function of finite order with zeros which are all real but neither bounded above nor bounded below, such that $$E'(z) = \pm 1$$ E ′ ( z ) = ± 1 whenever $$E(z) = 0$$ E ( z ) = 0 . Then either E has an explicit representation in terms of trigonometric functions or the zeros of E have exponent of convergence at least 3. An example constructed via quasiconformal surgery demonstrates the sharpness of this result.

scholarly journals On entire solutions of a certain type of nonlinear differential equation

2001 ◽  
Vol 64 (3) ◽  
pp. 377-380 ◽  
Author(s):  
Chung-Chun Yang
Keyword(s):  
Differential Equation ◽  
Entire Function ◽  
Finite Order ◽  
Nonlinear Differential Equation ◽  
Entire Solution ◽  
Differential Polynomial ◽  
Entire Solutions ◽  
Value Distribution Theory ◽  
Linear Differential Polynomial ◽  
Linear Differential

In this note, we shall study, via Nevanlinna's value distribution theory, the uniqueness of transcendental entire solutions of the following type of nonlinear differential equation: (*) L (f (z)) – p (z) fn(z) = h (z), where L (f) denotes a linear differential polynomial in f with polynomials as its co-efficients, p (z) a polynomial (≢ 0), h an entire function, and n an integer ≥ 3. We show that if the equation (*) has a finite order transcendental entire solution, then it must be unique, unless L (f) ≡ 0.

scholarly journals Lebesgue measure of escaping sets of entire functions

2018 ◽  
Vol 40 (1) ◽  
pp. 89-116 ◽  
Author(s):  
WEIWEI CUI
Keyword(s):  
Entire Function ◽  
Recent Work ◽  
Lebesgue Measure ◽  
Finite Order ◽  
Infinite Order ◽  
Entire Functions ◽  
Transcendental Entire Function ◽  
Positive Lebesgue Measure

For a transcendental entire function $f$ of finite order in the Eremenko–Lyubich class ${\mathcal{B}}$, we give conditions under which the Lebesgue measure of the escaping set ${\mathcal{I}}(f)$ of $f$ is zero. This complements the recent work of Aspenberg and Bergweiler [Math. Ann. 352(1) (2012), 27–54], in which they give conditions on entire functions in the same class with escaping sets of positive Lebesgue measure. We will construct an entire function in the Eremenko–Lyubich class to show that the condition given by Aspenberg and Bergweiler is essentially sharp. Furthermore, we adapt our idea of proof to certain infinite-order entire functions. Under some restrictions to the growth of these entire functions, we show that the escaping sets have zero Lebesgue measure. This generalizes a result of Eremenko and Lyubich.

Set of defect values of an entire function of finite order

1988 ◽  
Vol 39 (3) ◽  
pp. 225-228
Author(s):  
A. �. Eremenko
Keyword(s):  
Entire Function ◽  
Finite Order
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