Differential equations related to family $\mathscr{A}$
Bull. Korean Math. Soc. 2011 Vol. 48, No. 2, 247-260
https://doi.org/10.4134/BKMS.2011.48.2.247
Printed March 1, 2011
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
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