scholarly journals Radial distributions of Julia sets of difference operators of entire solutions of complex differential equations

2022 ◽  
Vol 7 (4) ◽  
pp. 5133-5145
Author(s):  
Jingjing Li ◽  
◽  
Zhigang Huang
Keyword(s):  
Differential Equation ◽  
Entire Function ◽  
Lower Bound ◽  
Julia Sets ◽  
Differential Polynomial ◽  
Difference Operators ◽  
Transcendental Entire Function ◽  
Entire Solutions ◽  
Complex Differential Equation ◽  
Complex Differential Equations

<abstract><p>In this paper, we mainly investigate the radial distribution of Julia sets of difference operators of entire solutions of complex differential equation $ F(z)f^{n}(z)+P(z, f) = 0 $, where $ F(z) $ is a transcendental entire function and $ P(z, f) $ is a differential polynomial in $ f $ and its derivatives. We obtain that the set of common limiting directions of Julia sets of non-trivial entire solutions, their shifts have a definite range of measure. Moreover, an estimate of lower bound of measure of the set of limiting directions of Jackson difference operators of non-trivial entire solutions is given.</p></abstract>

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 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].

On The Zeros Of Solutions Of Second-Order Linear Differential Equations

1956 ◽  
Vol 8 ◽  
pp. 504-515 ◽  
Author(s):  
P. R. Beesack ◽  
Binyamin Schwarz
Keyword(s):  
Differential Equation ◽  
Lower Bound ◽  
Euclidean Distance ◽  
Regular Function ◽  
Zero Solution ◽  
Open Unit ◽  
Complex Differential Equation ◽  
Zeros Of Solutions ◽  
Linear Differential ◽  
Image Position

Introduction. In §1 of this paper we consider the complex differential equation1, |z| < 1,where q(z) is a regular function in the open unit circle. We shall give a lower bound for the non-Euclidean distance of any pair of zeros of any non-trivial (i.e., not identically zero) solution u(z) of (1).

Entire solutions of a variation of the eikonal equation and related PDEs

2020 ◽  
Vol 63 (3) ◽  
pp. 697-708
Author(s):  
Feng Lü
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Entire Function ◽  
Eikonal Equation ◽  
Entire Solutions ◽  
Partial Differential ◽  
Tubular Surfaces

AbstractThe aim of this paper is twofold. The first aim is to describe the entire solutions of the partial differential equation (PDE) $u_{z_1}^2+2Bu_{z_1}u_{z_2}+u_{z_2}^2=e^g$, where B is a constant and g is a polynomial or an entire function in $\mathbb {C}^2$. The second aim is to consider the entire solutions of another PDE, which is a generalization of the well-known PDE of tubular surfaces.

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.

The Hausdorff dimension of Julia sets of entire functions III

1997 ◽  
Vol 122 (2) ◽  
pp. 223-244 ◽  
Author(s):  
GWYNETH M. STALLARD
Keyword(s):  
Entire Function ◽  
Hausdorff Dimension ◽  
Entire Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Transcendental Entire Function ◽  
Transcendental Entire Functions

It is known that, for a transcendental entire function f, the Hausdorff dimension of the Julia set of f satisfies 1[les ]dim J(f)[les ]2. In this paper we introduce a family of transcendental entire functions fp, K for which the set {dim J(fp, K)[ratio ]0<p, K<∞} has infemum 1 and supremum 2.

Further study on the Brück conjecture and some non-linear complex differential equations

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Dilip Chandra Pramanik ◽  
Kapil Roy
Keyword(s):  
Differential Equations ◽  
Current Paper ◽  
Small Function ◽  
Entire Solutions ◽  
Linear Complex ◽  
Content Type ◽  
Complex Differential Equation ◽  
Non Linear ◽  
Complex Differential Equations ◽  
Brück Conjecture

PurposeThe purpose of this current paper is to deal with the study of non-constant entire solutions of some non-linear complex differential equations in connection to Brück conjecture, by using the theory of complex differential equation. The results generalize the results due to Pramanik et al.Design/methodology/approach39B32, 30D35.FindingsIn the current paper, we mainly study the Brück conjecture and the various works that confirm this conjecture. In our study we find that the conjecture can be generalized for differential monomials under some additional conditions and it generalizes some works related to the conjecture. Also we can take the complex number a in the conjecture to be a small function. More precisely, we obtain a result which can be restate in the following way: Let f be a non-constant entire function such that σ2(f)<∞, σ2(f) is not a positive integer and δ(0, f)>0. Let M[f] be a differential monomial of f of degree γM and α(z), β(z)∈S(f) be such that max{σ(α), σ(β)} <σ(f). If M[f]+β and fγM−α share the value 0 CM, then M[f]+βfγM−α=c,where c≠0 is a constant.Originality/valueThis is an original work of the authors.

Entire function sharing a small function with its mixed-operators

2019 ◽  
Vol 26 (1) ◽  
pp. 47-62 ◽  
Author(s):  
Xianjing Dong ◽  
Kai Liu
Keyword(s):  
Entire Function ◽  
Linear Combination ◽  
Small Function ◽  
Uniqueness Problem ◽  
Difference Operators ◽  
Transcendental Entire Function ◽  
Shift Operators ◽  
Function Sharing

Abstract In this article, we investigate the uniqueness problem on a transcendental entire function {f(z)} with its linear mixed-operators Tf, where T is a linear combination of differential-difference operators {D^{\nu}_{\eta}:=f^{(\nu)}(z+\eta)} and shift operators {E_{\zeta}:=f(z+\zeta\/)} , where {\eta,\nu,\zeta} are constants. We obtain that if a transcendental entire function {f(z)} satisfies {\lambda(f-\alpha)<\sigma(f\/)<+\infty} , where {\alpha(z)} is an entire function with {\sigma(\alpha)<1} , and if f and Tf share one small entire function {a(z)} with {\sigma(a)<\sigma(f\/)} , then {\frac{Tf-a(z)}{f(z)-a(z)}=\tau,} where τ is a non-zero constant. Furthermore, we obtain the value τ and the expression of f by imposing additional conditions.

scholarly journals Spiders’ webs and locally connected Julia sets of transcendental entire functions

2012 ◽  
Vol 33 (4) ◽  
pp. 1146-1161 ◽  
Author(s):  
J. W. OSBORNE
Keyword(s):  
Entire Function ◽  
Opposite Direction ◽  
Entire Functions ◽  
Dense Subset ◽  
Julia Set ◽  
Julia Sets ◽  
Transcendental Entire Function ◽  
Transcendental Entire Functions

AbstractWe show that if the Julia set of a transcendental entire function is locally connected, then it takes the form of a spider’s web in the sense defined by Rippon and Stallard. In the opposite direction, we prove that a spider’s web Julia set is always locally connected at a dense subset of buried points. We also show that the set of buried points (the residual Julia set) can be a spider’s web.

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