scholarly journals Relationship between entire functions and their derivatives sharing small function except a set

Filomat ◽  
2018 ◽  
Vol 32 (19) ◽  
pp. 6845-6855
Author(s):  
Feng Lü
Keyword(s):  
Differential Equation ◽  
Entire Function ◽  
Entire Functions ◽  
Infinite Product ◽  
Small Function ◽  
Entire Solutions ◽  
Infinite Products ◽  
Growth Property ◽  
The Relationship

The paper is mainly devoted to deriving the relationship between an entire function and its derivative when they share one small function except possibly a set, which is related to the famous Br?ck conjecture. In addition, two propositions of infinite products are obtained. The first one is the growth property of a certain infinite product. The second one is the property of entire solutions of the differential equation which concerns infinite products.

A note on meromorphic functions with finite order and of bounded l-index

2021 ◽  
Vol 18 (1) ◽  
pp. 1-11
Author(s):  
Andriy Bandura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Finite Order ◽  
General Problem ◽  
Meromorphic Functions ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Meromorphic Solutions ◽  
Entire Solutions ◽  
Riccati Differential Equation

We present a generalization of concept of bounded $l$-index for meromorphic functions of finite order. Using known results for entire functions of bounded $l$-index we obtain similar propositions for meromorphic functions. There are presented analogs of Hayman's theorem and logarithmic criterion for this class. The propositions are widely used to investigate $l$-index boundedness of entire solutions of differential equations. Taking this into account we raise a general problem of generalization of some results from theory of entire functions of bounded $l$-index by meromorphic functions of finite order and their applications to meromorphic solutions of differential equations. There are deduced sufficient conditions providing $l$-index boundedness of meromoprhic solutions of finite order for the Riccati differential equation. Also we proved that the Weierstrass $\wp$-function has bounded $l$-index with $l(z)=|z|.$

scholarly journals Relation between Small Functions with Differential Polynomials Generated by Solutions of Linear Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Zhigang Huang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Entire Functions ◽  
Linear Differential Equations ◽  
Differential Polynomials ◽  
Small Functions ◽  
Linear Differential ◽  
The Relationship ◽  
Growth Of Solutions

This paper is devoted to studying the growth of solutions of second-order nonhomogeneous linear differential equation with meromorphic coefficients. We also discuss the relationship between small functions and differential polynomialsL(f)=d2f″+d1f′+d0fgenerated by solutions of the above equation, whered0(z),d1(z),andd2(z)are entire functions that are not all equal to zero.

scholarly journals A Class of functional equations which have entire solutions

1988 ◽  
Vol 38 (3) ◽  
pp. 351-356 ◽  
Author(s):  
Peter L. Walker
Keyword(s):  
Functional Equation ◽  
Entire Function ◽  
Wide Class ◽  
Functional Equations ◽  
Entire Functions ◽  
Inverse Function ◽  
Entire Solution ◽  
Entire Solutions ◽  
Image Position

We consider the Abelian functional equationwhere φ is a given entire function and g is to be found. The inverse function f = g−1 (if one exists) must satisfyWe show that for a wide class of entire functions, which includes φ(z) = ez − 1, the latter equation has a non-constant entire solution.

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 Entire functions that share a small function with their linear difference polynomial

2021 ◽  
Vol 7 (3) ◽  
pp. 3731-3744
Author(s):  
Minghui Zhange ◽  
◽  
Jianbin Xiao ◽  
Mingliang Fang
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Small Function ◽  
Difference Polynomial ◽  
Function Sharing

<abstract><p>In this paper, we investigate the uniqueness of an entire function sharing a small function with its linear difference polynomial. Our results improve some results due to Li and Yi <sup>[<xref ref-type="bibr" rid="b11">11</xref>]</sup>, Zhang, Chen and Huang <sup>[<xref ref-type="bibr" rid="b17">17</xref>]</sup>, Zhang, Kang and Liao <sup>[<xref ref-type="bibr" rid="b18">18</xref>,<xref ref-type="bibr" rid="b19">19</xref>]</sup> etc.</p></abstract>

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.

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>

ON THE BRÜCK CONJECTURE

2015 ◽  
Vol 93 (2) ◽  
pp. 248-259 ◽  
Author(s):  
TINGBIN CAO
Keyword(s):  
Entire Function ◽  
Differential Equations ◽  
Entire Functions ◽  
Linear Differential Equations ◽  
Small Function ◽  
Shared Value ◽  
Nonzero Constant ◽  
Hyper Order ◽  
Linear Differential ◽  
Brück Conjecture

The Brück conjecture states that if a nonconstant entire function $f$ with hyper-order ${\it\sigma}_{2}(f)\in [0,+\infty )\setminus \mathbb{N}$ shares one finite value $a$ (counting multiplicities) with its derivative $f^{\prime }$, then $f^{\prime }-a=c(f-a)$, where $c$ is a nonzero constant. The conjecture has been established for entire functions with order ${\it\sigma}(f)<+\infty$ and hyper-order ${\it\sigma}_{2}(f)<{\textstyle \frac{1}{2}}$. The purpose of this paper is to prove the Brück conjecture for the case ${\it\sigma}_{2}(f)=\frac{1}{2}$ by studying the infinite hyper-order solutions of the linear differential equations $f^{(k)}+A(z)f=Q(z)$. The shared value $a$ is extended to be a ‘small’ function with respect to the entire function $f$.

scholarly journals On the Transcendental Entire Solutions of a Certain Differential Equation and Fixed Points

2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Tiantian Shen ◽  
Weiran Lü ◽  
Guangxue Chu
Keyword(s):  
Differential Equation ◽  
Entire Function ◽  
Fixed Points ◽  
Entire Solutions ◽  
Open Question

We will first show that the following differential equation F(k)−z=eα(F−z) has transcendental entire solutions, where F=fn and α is an entire function, which improve what we have known and answer an open question raised in the work of Zhang and Yang (2009). And then, the examples are discussed.

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