scholarly journals Fast escaping points of entire functions: a new regularity condition

2015 ◽  
Vol 160 (1) ◽  
pp. 95-106
Author(s):  
V. EVDORIDOU
Keyword(s):  
Entire Function ◽  
Finite Order ◽  
Lower Order ◽  
Regularity Condition ◽  
Entire Functions ◽  
Transcendental Entire Function ◽  
Transcendental Dynamics

AbstractLet f be a transcendental entire function. The fast escaping set, A(f), plays a key role in transcendental dynamics. The quite fast escaping set, Q(f), defined by an apparently weaker condition is equal to A(f) under certain conditions. Here we introduce Q2(f) defined by what appears to be an even weaker condition. Using a new regularity condition we show that functions of finite order and positive lower order satisfy Q2(f) = A(f). We also show that the finite composition of such functions satisfies Q2(f) = A(f). Finally, we construct a function for which Q2(f) ≠ Q(f) = A(f).

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.

Algebraic Values of Entire Functions with Extremal Growth Orders: An Extension of a Theorem of Boxall and Jones

2019 ◽  
Vol 63 (3) ◽  
pp. 536-546
Author(s):  
Taboka Prince Chalebgwa
Keyword(s):  
Entire Function ◽  
Finite Order ◽  
Lower Order ◽  
Entire Functions ◽  
Compact Sets ◽  
Bounded Degree ◽  
Growth Of Entire Functions ◽  
Original Theorem

AbstractGiven an entire function $f$ of finite order $\unicode[STIX]{x1D70C}$ and positive lower order $\unicode[STIX]{x1D706}$, Boxall and Jones proved a bound of the form $C(\log H)^{\unicode[STIX]{x1D702}(\unicode[STIX]{x1D706},\unicode[STIX]{x1D70C})}$ for the density of algebraic points of bounded degree and height at most $H$ on the restrictions to compact sets of the graph of $f$. The constant $C$ and exponent $\unicode[STIX]{x1D702}$ are effectively computable from certain data associated with the function. In this followup note, using different measures of the growth of entire functions, we obtain similar bounds for other classes of functions to which the original theorem does not apply.

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 Fatou’s Associates

2020 ◽  
Vol 6 (3-4) ◽  
pp. 459-493
Author(s):  
Vasiliki Evdoridou ◽  
Lasse Rempe ◽  
David J. Sixsmith
Keyword(s):  
Entire Function ◽  
Blaschke Product ◽  
Entire Functions ◽  
Julia Set ◽  
Singular Values ◽  
Strong Relationship ◽  
Inner Function ◽  
Transcendental Entire Function ◽  
Fatou Component ◽  
Finite Blaschke Product

AbstractSuppose that f is a transcendental entire function, $$V \subsetneq {\mathbb {C}}$$ V ⊊ C is a simply connected domain, and U is a connected component of $$f^{-1}(V)$$ f - 1 ( V ) . Using Riemann maps, we associate the map $$f :U \rightarrow V$$ f : U → V to an inner function $$g :{\mathbb {D}}\rightarrow {\mathbb {D}}$$ g : D → D . It is straightforward to see that g is either a finite Blaschke product, or, with an appropriate normalisation, can be taken to be an infinite Blaschke product. We show that when the singular values of f in V lie away from the boundary, there is a strong relationship between singularities of g and accesses to infinity in U. In the case where U is a forward-invariant Fatou component of f, this leads to a very significant generalisation of earlier results on the number of singularities of the map g. If U is a forward-invariant Fatou component of f there are currently very few examples where the relationship between the pair (f, U) and the function g has been calculated. We study this relationship for several well-known families of transcendental entire functions. It is also natural to ask which finite Blaschke products can arise in this manner, and we show the following: for every finite Blaschke product g whose Julia set coincides with the unit circle, there exists a transcendental entire function f with an invariant Fatou component such that g is associated with f in the above sense. Furthermore, there exists a single transcendental entire function f with the property that any finite Blaschke product can be arbitrarily closely approximated by an inner function associated with the restriction of f to a wandering domain.

Entire functions of slow growth whose Julia set coincides with the plane

2000 ◽  
Vol 20 (6) ◽  
pp. 1577-1582 ◽  
Author(s):  
WALTER BERGWEILER ◽  
ALEXANDRE EREMENKO
Keyword(s):  
Entire Function ◽  
Slow Growth ◽  
Entire Functions ◽  
Julia Set ◽  
Transcendental Entire Function

We construct a transcendental entire function $f$ with $J(f)=\mathbb{C}$ such that $f$ has arbitrarily slow growth; that is, $\log |f(z)|\leq\phi(|z|)\log |z|$ for $|z|>r_0$, where $\phi$ is an arbitrary prescribed function tending to infinity.

scholarly journals On relative (p,q)-th order based growth measure of entire functions

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1723-1735
Author(s):  
Sanjib Datta ◽  
Tanmay Biswas ◽  
Chinmay Ghosh
Keyword(s):  
Entire Function ◽  
Lower Order ◽  
Entire Functions ◽  
Positive Integers

In this paper we intend to find out relative (p,q)-th order (relative (p,q)-th lower order) of an entire function f with respect to another entire function 1 when relative (m,q)-th order (relative (m,q)-th lower order) of f and relative (m,p)-th order (relative (m,p)-th lower order) of g with respect to another entire function h are given with p,q,m are all positive integers.

scholarly journals Entire functions mapping countable dense subsets of the reals onto each other monotonically

1974 ◽  
Vol 10 (1) ◽  
pp. 67-70 ◽  
Author(s):  
Daihachiro Sato ◽  
Stuart Rankin
Keyword(s):  
Entire Function ◽  
Real Line ◽  
Entire Functions ◽  
Transcendental Entire Function ◽  
The Real

It is shown that for arbitrary countable dense ssets A and B of the real line, there exists a transcendental entire function whose restriction to the real line is a real-valued strictly monotone increasing surjection taking A onto B The technique used is a modification of the procedure Maurer used to show that for countable dense subsets A and B of the plane, there exists a transcendental entire function whose restriction to A is a bijection from A to B.

scholarly journals On the value distribution of iterated entire functions

1999 ◽  
Vol 66 (1) ◽  
pp. 18-31 ◽  
Author(s):  
Zheng Jian-Hua
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Periodic Points ◽  
Value Distribution ◽  
Transcendental Entire Function

AbstractLet f be a transcendental entire function and denote the n-th iterate fn. For n ≥ 2, we give an explict estimate of the number of periodic points of f with period n, that is, fix-points of fn which are not fix-points of fk for 1 ≤ k <n.

scholarly journals Some unity results on entire functions and their difference operators related to 4 CM theorem

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
BaoQin Chen ◽  
Sheng Li
Keyword(s):  
Entire Function ◽  
Finite Order ◽  
Entire Functions ◽  
Difference Operators ◽  
Finite Complex

Abstract This paper is to consider the unity results on entire functions sharing two values with their difference operators and to prove some results related to 4 CM theorem. The main result reads as follows: Let $f(z)$ f ( z ) be a nonconstant entire function of finite order, and let $a_{1}$ a 1 , $a_{2}$ a 2 be two distinct finite complex constants. If $f(z)$ f ( z ) and $\Delta _{\eta }^{n}f(z)$ Δ η n f ( z ) share $a_{1}$ a 1 and $a_{2}$ a 2 “CM”, then $f(z)\equiv \Delta _{\eta }^{n} f(z)$ f ( z ) ≡ Δ η n f ( z ) , and hence $f(z)$ f ( z ) and $\Delta _{\eta }^{n}f(z)$ Δ η n f ( z ) share $a_{1}$ a 1 and $a_{2}$ a 2 CM.

The Hausdorff dimension of Julia sets of entire functions II

1996 ◽  
Vol 119 (3) ◽  
pp. 513-536 ◽  
Author(s):  
Gwyneth M. Stallard
Keyword(s):  
Entire Function ◽  
Hausdorff Dimension ◽  
Entire Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Transcendental Entire Function ◽  
Bounded Set

AbstractLetfbe a transcendental entire function such that the finite singularities of f−1lie in a bounded set. We show that the Hausdorff dimension of the Julia set of such a function is strictly greater than one.

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