scholarly journals Uniformly bounded components of normality

2007 ◽  
Vol 143 (1) ◽  
pp. 85-101
Author(s):  
XIAOLING WANG ◽  
WANG ZHOU
Keyword(s):  
Entire Function ◽  
Transcendental Entire Function ◽  
Fatou Set ◽  
Image Position ◽  
Uniformly Bounded

AbstractSuppose that f(z) is a transcendental entire function and that the Fatou set F(f)≠∅. Set and where the supremum supU is taken over all components of F(f). If B1(f)<∞ or B2(f)<∞, then we say F(f) is strongly uniformly bounded or uniformly bounded respectively. We show that, under some conditions, F(f) is (strongly) uniformly bounded.

scholarly journals ON SOLUTIONS TO NONHOMOGENEOUS ALGEBRAIC DIFFERENTIAL EQUATIONS AND THEIR APPLICATION

2014 ◽  
Vol 97 (3) ◽  
pp. 391-403 ◽  
Author(s):  
LIANG-WEN LIAO ◽  
ZHUAN YE
Keyword(s):  
Differential Equation ◽  
Entire Function ◽  
Rational Function ◽  
Meromorphic Functions ◽  
Meromorphic Solution ◽  
Differential Polynomial ◽  
Transcendental Entire Function ◽  
Algebraic Differential Equation ◽  
Algebraic Differential Equations ◽  
Image Position

AbstractWe consider solutions to the algebraic differential equation $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f^nf'+Q_d(z,f)=u(z)e^{v(z)}$, where $Q_d(z,f)$ is a differential polynomial in $f$ of degree $d$ with rational function coefficients, $u$ is a nonzero rational function and $v$ is a nonconstant polynomial. In this paper, we prove that if $n\ge d+1$ and if it admits a meromorphic solution $f$ with finitely many poles, then $$\begin{equation*} f(z)=s(z)e^{v(z)/(n+1)} \quad \mbox {and}\quad Q_d(z,f)\equiv 0. \end{equation*}$$ With this in hand, we also prove that if $f$ is a transcendental entire function, then $f'p_k(f)+q_m(f)$ assumes every complex number $\alpha $, with one possible exception, infinitely many times, where $p_k(f), q_m(f)$ are polynomials in $f$ with degrees $k$ and $m$ with $k\ge m+1$. This result generalizes a theorem originating from Hayman [‘Picard values of meromorphic functions and their derivatives’, Ann. of Math. (2)70(2) (1959), 9–42].

scholarly journals Structurally Similar Sets of Transcendental Entire Function

2018 ◽  
Vol 4 ◽  
pp. 11-16
Author(s):  
Bishnu Hari Subedi
Keyword(s):  
Entire Function ◽  
Complex Plane ◽  
Full Text ◽  
Complex Dynamics ◽  
Julia Set ◽  
Transcendental Entire Function ◽  
Fatou Set ◽  
Classical Sense

In complex dynamics, the complex plane is partitioned into invariant subsets. In classical sense, these subsets are of course Fatou set and Julia set. Rest of the abstract available with the full text

A Quantitative Estimate on Fixed-Points of Composite Meromorphic Functions

1995 ◽  
Vol 38 (4) ◽  
pp. 490-495 ◽  
Author(s):  
Jian-Hua Zheng
Keyword(s):  
Entire Function ◽  
Meromorphic Function ◽  
Fixed Points ◽  
Lower Order ◽  
Quantitative Estimate ◽  
Meromorphic Functions ◽  
Transcendental Entire Function ◽  
Logarithmic Density ◽  
Image Position ◽  
Finite Lower Order

AbstractLet ƒ(z) be a transcendental meromorphic function of finite order, g(z) a transcendental entire function of finite lower order and let α(z) be a non-constant meromorphic function with T(r, α) = S(r,g). As an extension of the main result of [7], we prove thatwhere J has a positive lower logarithmic density.

scholarly journals Periodic domains of quasiregular maps

2017 ◽  
Vol 38 (6) ◽  
pp. 2321-2344 ◽  
Author(s):  
DANIEL A. NICKS ◽  
DAVID J. SIXSMITH
Keyword(s):  
Entire Function ◽  
Half Space ◽  
General Result ◽  
Periodic Component ◽  
Transcendental Entire Function ◽  
Periodic Domain ◽  
Fatou Set ◽  
Additional Hypothesis ◽  
Lipschitz Maps ◽  
Periodic Domains

We consider the iteration of quasiregular maps of transcendental type from $\mathbb{R}^{d}$ to $\mathbb{R}^{d}$. We give a bound on the rate at which the iterates of such a map can escape to infinity in a periodic component of the quasi-Fatou set. We give examples which show that this result is the best possible. Under an additional hypothesis, which is satisfied by all uniformly quasiregular maps, this bound can be improved to be the same as those in a Baker domain of a transcendental entire function. We construct a quasiregular map of transcendental type from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ with a periodic domain in which all iterates tend locally uniformly to infinity. This is the first example of such behaviour in a dimension greater than two. Our construction uses a general result regarding the extension of bi-Lipschitz maps. In addition, we show that there is a quasiregular map of transcendental type from $\mathbb{R}^{3}$ to $\mathbb{R}^{3}$ which is equal to the identity map in a half-space.

scholarly journals Oscillation results on y″ + Ay = 0 in the complex domain with transcendental entire coefficients which have extremal deficiencies

1995 ◽  
Vol 38 (1) ◽  
pp. 13-34 ◽  
Author(s):  
Y. M. Chiang
Keyword(s):  
Entire Function ◽  
Complex Domain ◽  
Transcendental Entire Function ◽  
Exponent Of Convergence ◽  
Linearly Independent ◽  
Image Position ◽  
Nevanlinna Deficiency

Let A(z) be a transcendental entire function and f1, f2 be linearly independent solutions ofWe prove that if A(z) has Nevanlinna deficiency δ(0, A) = 1, then the exponent of convergence of E: = flf2 is infinite. The theorems that we prove here are similar to those in Bank, Laine and Langley [3].

scholarly journals On the oscillation of solutions of certain linear differential equations in the complex domain

1987 ◽  
Vol 30 (3) ◽  
pp. 455-469 ◽  
Author(s):  
Steven B. Bank ◽  
J. K. Langley
Keyword(s):  
Differential Equation ◽  
Entire Function ◽  
Trivial Solution ◽  
Infinite Order ◽  
Finite Measure ◽  
Transcendental Entire Function ◽  
Starting Point ◽  
Linear Differential ◽  
Image Position ◽  
Oscillation Of Solutions

Our starting point is the differential equationwhere A(z) is a transcendental entire function of finite order, and we are concerned specifically with the frequency of zeros of a non-trivial solution f(z) of (1.1). Of course it is well known that such a solution f(z) is an entire function of infinite order, and using standard notation from [7],for all , b∈C\{0}, at least outside a set of r of finite measure.

scholarly journals Fatou components of entire functions of small growth

1999 ◽  
Vol 19 (5) ◽  
pp. 1281-1293 ◽  
Author(s):  
XINHOU HUA ◽  
CHUNG-CHUN YANG
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Transcendental Entire Function ◽  
Fatou Set ◽  
Transcendental Entire Functions ◽  
Small Growth ◽  
Fatou Components

This paper is concerned with the dynamics of transcendental entire functions. Let $f(z)$ be a transcendental entire function. We shall study the boundedness of the components of the Fatou set $F(f)$ under some restrictions on the growth of the function. This relates to a problem due to Baker in 1981.

On uniformly perfect boundary of stable domains in iteration of meromorphic functions II

2002 ◽  
Vol 132 (3) ◽  
pp. 531-544 ◽  
Author(s):  
ZHENG JIAN-HUA
Keyword(s):  
Entire Function ◽  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Fatou Set ◽  
Deficient Value ◽  
Uniformly Perfect ◽  
Simply Connected ◽  
Stable Domains ◽  
Uniform Perfectness

We investigate uniform perfectness of the Julia set of a transcendental meromorphic function with finitely many poles and prove that the Julia set of such a meromorphic function is not uniformly perfect if it has only bounded components. The Julia set of an entire function is uniformly perfect if and only if the Julia set including infinity is connected and every component of the Fatou set is simply connected. Furthermore if an entire function has a finite deficient value in the sense of Nevanlinna, then it has no multiply connected stable domains. Finally, we give some examples of meromorphic functions with uniformly perfect Julia sets.

scholarly journals Singular perturbations of the unicritical polynomials with two parameters

2016 ◽  
Vol 37 (6) ◽  
pp. 1997-2016 ◽  
Author(s):  
YINGQING XIAO ◽  
FEI YANG
Keyword(s):  
Julia Set ◽  
Dynamical Behavior ◽  
Rational Maps ◽  
Topological Properties ◽  
Fatou Set ◽  
The Family ◽  
Image Position ◽  
Two Parameters ◽  
Unicritical Polynomials

In this paper, we study the dynamics of the family of rational maps with two parameters $$\begin{eqnarray}f_{a,b}(z)=z^{n}+\frac{a^{2}}{z^{n}-b}+\frac{a^{2}}{b},\end{eqnarray}$$ where $n\geq 2$ and $a,b\in \mathbb{C}^{\ast }$. We give a characterization of the topological properties of the Julia set and the Fatou set of $f_{a,b}$ according to the dynamical behavior of the orbits of the free critical points.

On the Properties of an Entire Function of Two Complex Variables

1968 ◽  
Vol 20 ◽  
pp. 51-57
Author(s):  
Arun Kumar Agarwal
Keyword(s):  
Entire Function ◽  
Double Series ◽  
Complex Variables ◽  
Maximum Term ◽  
Image Position

1. Letbe an entire function of two complex variables z1 and z2, holomorphic in the closed polydisk . LetFollowing S. K. Bose (1, pp. 214-215), μ(r1, r2; ƒ ) denotes the maximum term in the double series (1.1) for given values of r1 and r2 and v1{m2; r1, r2) or v1(r1, r2), r2 fixed, v2(m1, r1, r2) or v2(r1, r2), r1 fixed and v(r1r2) denote the ranks of the maximum term of the double series (1.1).

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