scholarly journals Uniqueness of Entire Functions Sharing Polynomials with Their Derivatives

2009 ◽  
Vol 2009 ◽  
pp. 1-9
Author(s):  
Jianming Qi ◽  
Feng Lü ◽  
Ang Chen
Keyword(s):  
Entire Function ◽  
Nonnegative Integer ◽  
Entire Functions ◽  
Normal Families ◽  
Transcendental Entire Function ◽  
Complex Numbers

We use the theory of normal families to prove the following. LetQ1(z)=a1zp+a1,p−1zp−1+⋯+a1,0andQ2(z)=a2zp+a2,p−1zp−1+⋯+a2,0be two polynomials such thatdeg⁡Q1=deg⁡Q2=p(wherepis a nonnegative integer) anda1,a2(a2≠0)are two distinct complex numbers. Letf(z)be a transcendental entire function. Iff(z)andf′(z)share the polynomialQ1(z) CMand iff(z)=Q2(z)wheneverf′(z)=Q2(z), thenf≡f′. This result improves a result due to Li and Yi.

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 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 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.

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.

scholarly journals Maximally and non-maximally fast escaping points of transcendental entire functions

2015 ◽  
Vol 158 (2) ◽  
pp. 365-383 ◽  
Author(s):  
D. J. SIXSMITH
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Julia Set ◽  
Transcendental Entire Function ◽  
Transcendental Entire Functions ◽  
Dynamical Properties

AbstractWe partition the fast escaping set of a transcendental entire function into two subsets, the maximally fast escaping set and the non-maximally fast escaping set. These sets are shown to have strong dynamical properties. We show that the intersection of the Julia set with the non-maximally fast escaping set is never empty. The proof uses a new covering result for annuli, which is of wider interest.It was shown by Rippon and Stallard that the fast escaping set has no bounded components. In contrast, by studying a function considered by Hardy, we give an example of a transcendental entire function for which the maximally and non-maximally fast escaping sets each have uncountably many singleton components.

scholarly journals Dynamics of slowly growing entire functions

2001 ◽  
Vol 63 (3) ◽  
pp. 367-377 ◽  
Author(s):  
I. N. Baker
Keyword(s):  
Entire Function ◽  
Normal Family ◽  
Entire Functions ◽  
Julia Set ◽  
Maximum Modulus ◽  
Stable Set ◽  
Transcendental Entire Function ◽  
Transcendental Entire Functions ◽  
Transcendental Functions ◽  
Simply Connected

Dedicated to George Szekeres on his 90th birthdayFor a transcendental entire function f let M(r) denote the maximum modulus of f(z) for |z| = r. Then A(r) = log M(r)/logr tends to infinity with r. Many properties of transcendental entire functions with sufficiently small A(r) resemble those of polynomials. However the dynamical properties of iterates of such functions may be very different. For instance in the stable set F(f) where the iterates of f form a normal family the components are preperiodic under f in the case of a polynomial; but there are transcendental functions with arbitrarily small A(r) such that F(f) has nonpreperiodic components, so called wandering components, which are bounded rings in which the iterates tend to infinity. One might ask if all small functions are like this.A striking recent result of Bergweiler and Eremenko shows that there are arbitrarily small transcendental entire functions with empty stable set—a thing impossible for polynomials. By extending the technique of Bergweiler and Eremenko, an arbitrarily small transcendental entire function is constructed such that F is nonempty, every component G of F is bounded, simply-connected and the iterates tend to zero in G. Zero belongs to an invariant component of F, so there are no wandering components. The Julia set which is the complement of F is connected and contains a dense subset of “buried’ points which belong to the boundary of no component of F. This bevaviour is impossible for a polynomial.

scholarly journals Dynamical sets whose union with infinity is connected

2018 ◽  
Vol 40 (3) ◽  
pp. 789-798 ◽  
Author(s):  
DAVID J. SIXSMITH
Keyword(s):  
Entire Function ◽  
Large Class ◽  
Topological Structure ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Related Question ◽  
Transcendental Entire Function ◽  
Transcendental Entire Functions ◽  
Class Of Functions

Suppose that $f$ is a transcendental entire function. In 2011, Rippon and Stallard showed that the union of the escaping set with infinity is always connected. In this paper we consider the related question of whether the union with infinity of the bounded orbit set, or the bungee set, can also be connected. We give sufficient conditions for these sets to be connected and an example of a transcendental entire function for which all three sets are simultaneously connected. This function lies, in fact, in the Speiser class.It is known that for many transcendental entire functions the escaping set has a topological structure known as a spider’s web. We use our results to give a large class of functions in the Eremenko–Lyubich class for which the escaping set is not a spider’s web. Finally, we give a novel topological criterion for certain sets to be a spider’s web.

ON THE PERIODICITY OF TRANSCENDENTAL ENTIRE FUNCTIONS

2020 ◽  
Vol 101 (3) ◽  
pp. 453-465 ◽  
Author(s):  
XINLING LIU ◽  
RISTO KORHONEN
Keyword(s):  
Entire Function ◽  
Positive Integer ◽  
Periodic Function ◽  
Entire Functions ◽  
Transcendental Entire Function ◽  
Difference Polynomials ◽  
Transcendental Entire Functions ◽  
Hyper Order

According to a conjecture by Yang, if $f(z)f^{(k)}(z)$ is a periodic function, where $f(z)$ is a transcendental entire function and $k$ is a positive integer, then $f(z)$ is also a periodic function. We propose related questions, which can be viewed as difference or differential-difference versions of Yang’s conjecture. We consider the periodicity of a transcendental entire function $f(z)$ when differential, difference or differential-difference polynomials in $f(z)$ are periodic. For instance, we show that if $f(z)^{n}f(z+\unicode[STIX]{x1D702})$ is a periodic function with period $c$, then $f(z)$ is also a periodic function with period $(n+1)c$, where $f(z)$ is a transcendental entire function of hyper-order $\unicode[STIX]{x1D70C}_{2}(f)<1$ and $n\geq 2$ is an integer.

Sign in / Sign up

Export Citation Format

Share Document