Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2011; 48(2): 247-260

Printed March 1, 2011

https://doi.org/10.4134/BKMS.2011.48.2.247

Copyright © The Korean Mathematical Society.

Differential equations related to family $\mathscr{A}$

Ping Li and Yong Meng

University of Science and Technology of China, Hefei University of Technology

Abstract

Let $h$ be a meromorphic function with few poles and zeros. By Nevanlinna's value distribution theory we prove some new properties on the polynomials in $h$ with the coefficients being small functions of $h$. We prove that if $f$ is a meromorphic function and if $f^m$ is identically a polynomial in $h$ with the constant term not vanish identically, then $f$ is a polynomial in $h.$ As an application, we are able to find the entire solutions of the differential equation of the type $$f^n+P(f)=be^{sz}+Q(e^z),$$ where $P(f)$ is a differential polynomial in $f$ of degree at most $n-1,$ and $Q(e^z)$ is a polynomial in $e^z$ of degree $k\leqslant\max\{n-1,s(n-1)/n\}$ with small functions of $e^z$ as its coefficients.

Keywords: Nevanlinna theory, meromorphic function, differential equation

MSC numbers: 30D35