scholarly journals Growth of Meromorphic Function Sharing Functions and Some Uniqueness Problems

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Jianming Qi ◽  
Fanning Meng ◽  
Wenjun Yuan
Keyword(s):  
Meromorphic Function ◽  
Differential Equations ◽  
Meromorphic Functions ◽  
Meromorphic Solutions ◽  
Complex Differential Equations ◽  
Function Sharing

Estimating the growth of meromorphic solutions has been an important topic of research in complex differential equations. In this paper, we devoted to considering uniqueness problems by estimating the growth of meromorphic functions. Further, some examples are given to show that the conclusions are meaningful.

On some generalized Painlevé and Hayman type equations with meromorphic solutions in a bounded domain

2018 ◽  
Vol 25 (2) ◽  
pp. 187-194 ◽  
Author(s):  
Grigor Barsegian ◽  
Wenjun Yuan
Keyword(s):  
Differential Equations ◽  
Bounded Domain ◽  
Meromorphic Functions ◽  
Upper Bounds ◽  
Higher Order ◽  
Value Distribution ◽  
Meromorphic Solutions ◽  
Painleve Equations ◽  
Complex Differential Equations ◽  
Similar Solutions

Abstract The value distribution and, in particular, the numbers of a-points, have not been studied for meromorphic functions which are solutions of some complex differential equations in a given domain. Instead, the numbers of good a-points and Ahlfors islands, which play to a certain extend a role similar to that of the numbers of a-points, have been considered in some recent papers. In this paper, we consider meromorphic functions in a given domain, which are the solutions of some higher order equations and largely generalize the solutions of Painlevé equations 3–6. We give the upper bounds for the numbers of good a-points and Ahlfors islands of similar solutions.

scholarly journals Value Distribution of Meromorphic Solutions and Their Derivatives of Complex Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Abdallah El Farissi
Keyword(s):  
Differential Equations ◽  
Meromorphic Functions ◽  
Higher Order ◽  
Value Distribution ◽  
Meromorphic Solutions ◽  
Small Functions ◽  
Complex Differential Equations ◽  
Linear Differential ◽  
The Relationship ◽  
Derivatives Of

We deal with the relationship between the small functions and the derivatives of solutions of higher-order linear differential equations f(k)+Ak-1f(k-1)+⋯+A0f=0,   k≥2, where Aj(z)  (j=0,1,…,k-1) are meromorphic functions. The theorems of this paper improve the previous results given by El Farissi, Belaïdi, Wang, Lu, Liu, and Zhang.

scholarly journals On the Exact Solutions of Two (3+1)-Dimensional Nonlinear Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Fanning Meng ◽  
Yongyi Gu
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Shallow Water ◽  
Traveling Wave ◽  
Nonlinear Differential Equations ◽  
Complex Method ◽  
Meromorphic Solutions ◽  
Complex Differential Equations ◽  
Bkp Equation

In this article, exact solutions of two (3+1)-dimensional nonlinear differential equations are derived by using the complex method. We change the (3+1)-dimensional B-type Kadomtsev-Petviashvili (BKP) equation and generalized shallow water (gSW) equation into the complex differential equations by applying traveling wave transform and show that meromorphic solutions of these complex differential equations belong to class W, and then, we get exact solutions of these two (3+1)-dimensional equations.

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 On meromorphic solutions of certain nonlinear differential equations

2002 ◽  
Vol 66 (2) ◽  
pp. 331-343 ◽  
Author(s):  
J. Heittokangas ◽  
R. Korhonen ◽  
I. Laine
Keyword(s):  
Meromorphic Function ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Differential Polynomial ◽  
Meromorphic Solutions ◽  
Linear Differential Polynomial ◽  
Linear Differential

In this paper, we consider the growth of meromorphic solutions of nonlinear differential equations of the form L (f) + P (z, f) = h (z), where L (f) denotes a linear differential polynomial in f, P (z, f) is a polynomial in f, both with small meromorphic coefficients, and h (z) is a meromorphic function. Specialising to L (f) − p (z) fn = h (z), where p (z) is a small meromorphic function, we consider the uniqueness of meromorphic solutions with few poles only. Our results complement earlier ones due to C.-C. Yang.

scholarly journals On the growth of compositions of linear and meromorphic functions

1991 ◽  
Vol 44 (2) ◽  
pp. 263-269 ◽  
Author(s):  
Jianyong Qiao
Keyword(s):  
Meromorphic Function ◽  
Asymptotic Behaviour ◽  
Functional Equations ◽  
Meromorphic Functions ◽  
Meromorphic Solutions

Let f(z) be a meromorphic function; we shall investigate the asymptotic behaviour of the ratio T(r, f(z + α))/T(r, f(z)) and T(r, f(αz))/T(r, f(z)), and discuss the growth of the meromorphic solutions of some functional equations.

scholarly journals Some Properties on The [p,q]-Order of Meromorphic Solutions of Homogeneous and Non-homogeneous Linear Differential Equations With Meromorphic Coefficients

2021 ◽  
Vol 1 (2) ◽  
pp. 86-105
Author(s):  
Mansouria Saidani ◽  
Benharrat Belaidi
Keyword(s):  
Differential Equations ◽  
Meromorphic Functions ◽  
Higher Order ◽  
Linear Differential Equations ◽  
Meromorphic Solutions ◽  
Order Linear ◽  
Linear Differential ◽  
Right Order ◽  
Homogeneous Linear

In the present paper, we investigate the $\left[p,q\right] $-order of solutions of higher order linear differential equations \begin{equation*} A_{k}\left(z\right) f^{\left( k\right) }+A_{k-1}\left( z\right) f^{\left(k-1\right)}+\cdots +A_{1}\left( z\right) f^{\prime }+A_{0}\left( z\right)   f=0 \end{equation*} and \begin{equation*} A_{k}\left( z\right) f^{\left( k\right) }+A_{k-1}\left( z\right) f^{\left(k-1\right) }+\cdots +A_{1}\left( z\right) f^{\prime }+A_{0}\left( z\right) f=F\left( z\right), \end{equation*} where $A_{0}\left( z\right) ,$ $A_{1}\left( z\right) ,...,A_{k}\left(z\right) \not\equiv 0$ and $F\left( z\right) \not\equiv 0$ are meromorphic functions of finite $\left[ p,q\right] $-order. We improve and extend some results of the authors by using the concept $\left[ p,q\right] $-order.

scholarly journals The Oscillation on Solutions of Some Classes of Linear Differential Equations with Meromorphic Coefficients of Finite[p,q]-Order

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Hong-Yan Xu ◽  
Jin Tu ◽  
Zu-Xing Xuan
Keyword(s):  
Meromorphic Function ◽  
Differential Equations ◽  
Second Order ◽  
Linear Differential Equations ◽  
Meromorphic Solutions ◽  
Order Linear ◽  
Linear Differential

This paper considers the oscillation on meromorphic solutions of the second-order linear differential equations with the formf′′+A(z)f=0,whereA(z)is a meromorphic function with[p,q]-order. We obtain some theorems which are the improvement and generalization of the results given by Bank and Laine, Cao and Li, Kinnunen, and others.

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