On meromorphic solutions of non-linear differential equations of Tumura-Clunie type

Abstract

Meromorphic solutions of non-linear differential equations of the form $f^n+P(z,f)=h$ are investigated, where $n\ge 2$ is an integer, $h$ is a meromorphic function, and $P(z,f)$ is differential polynomial in $f$ and its derivatives with small functions as its coefficients. In the existing literature this equation has been studied in the case when $h$ has the particular form $h(z)=p_1(z)e^{{\alpha }_1(z)}+p_2(z)e^{{\alpha }_2(z)}$, where $p_1,p_2$ are small functions of $f$ and ${\alpha }_1,{\alpha }_2$ are entire functions. In such a case the order of $h$ is either a positive integer or equal to infinity. In this article it is assumed that $h$ is a meromorphic solution of the linear differential equation $h^{\prime \prime }+r_1(z)h^{\prime }+r_0(z)h=r_2(z)$ with rational coefficients $r_0,r_1,r_2$, and hence the order of $h$ is a rational number. Recent results by Liao-Yang-Zhang (2013) and Liao (2015) follow as special cases of the main results.

29 pages