On entire functions of infinite order

1963 ◽  
Vol 14 (1) ◽  
pp. 323-327 ◽  
Author(s):  
S. M. Shah
Keyword(s):  
Infinite Order ◽  
Entire Functions

scholarly journals On the Growth of Solutions of Some Second Order Linear Differential Equations with Entire Coefficients

2013 ◽  
Vol 21 (2) ◽  
pp. 35-52
Author(s):  
Benharrat Belaïdi ◽  
Habib Habib
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Infinite Order ◽  
Entire Functions ◽  
Complex Numbers ◽  
Order Of Growth ◽  
Hyper Order ◽  
Linear Differential ◽  
Growth Of Solutions

Abstract In this paper, we investigate the order and the hyper-order of growth of solutions of the linear differential equation where n≥2 is an integer, Aj (z) (≢0) (j = 1,2) are entire functions with max {σ A(j) : (j = 1,2} < 1, Q (z) = qmzm + ... + q1z + q0 is a nonoonstant polynomial and a1, a2 are complex numbers. Under some conditions, we prove that every solution f (z) ≢ 0 of the above equation is of infinite order and hyper-order 1.

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.

scholarly journals A landing theorem for entire functions with bounded post-singular sets

2020 ◽  
Vol 30 (6) ◽  
pp. 1465-1530
Author(s):  
Anna Miriam Benini ◽  
Lasse Rempe
Keyword(s):  
Entire Function ◽  
Periodic Point ◽  
Infinite Order ◽  
Entire Functions ◽  
Complex Polynomial ◽  
Landing Point ◽  
External Rays ◽  
Singular Sets ◽  
External Ray ◽  
Polynomial Dynamics

AbstractThe Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomial f with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray. We prove an analogue of this theorem for an entire function f with bounded postsingular set. If f has finite order of growth, then it is known that the escaping set I(f) contains certain curves called periodic hairs; we show that every periodic hair lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic hair. For a postsingularly bounded entire function f of infinite order, such hairs may not exist. Therefore we introduce certain dynamically natural connected subsets of I(f), called dreadlocks. We show that every periodic dreadlock lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic dreadlock. More generally, we prove that every point of a hyperbolic set is the landing point of a dreadlock.

Zeros of entire functions of infinite order

1992 ◽  
Vol 58 (4) ◽  
pp. 371-384
Author(s):  
A. M. Russakovskii
Keyword(s):  
Infinite Order ◽  
Entire Functions ◽  
Zeros Of Entire Functions

The Abel-Goncharov problem for entire functions of infinite order

1975 ◽  
Vol 16 (1) ◽  
pp. 59-67 ◽  
Author(s):  
V. A. Oskolkov
Keyword(s):  
Infinite Order ◽  
Entire Functions
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