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.