Meromorphic solutions to certain class of differential equations in an angular domain

Abstract

Abstract In this paper, we study the admissible meromorphic solutions to the algebraic differential equation fnf′+Pn−1(f)=uev $f^{n} f' + P_{n-1}( f ) = u\mathrm {e}^{v}$ in an angular domain instead of the whole complex plane, where Pn−1(f) $P_{n-1}(f)$ is a differential polynomial in f of degree ≤n−1 $\leq n-1$ with small function coefficients, u is a non-vanishing small function of f and v is an entire function. Herein, mainly, we are able to show that the equation does not admit any meromorphic solution f under some conditions unless Pn−1(f)≡0 $P_{n-1}(f)\equiv0$. Using this result, we are able to extend or generalize a well-known result of Hayman.

Keywords

• Nevanlinna theory on an angular domain
• Admissible solution
• Entire function
• Differential equation