On meromorphic solutions of functional-differential equations

Abstract

AbstractWe consider meromorphic solutions of functional-differential equations \[ f^{(k)}(z)=a(f^{n}\circ g)(z)+bf(z)+c, \]where $n,\,~k$ are two positive integers. Firstly, using an elementary method, we describe the forms of $f$ and $g$ when $f$ is rational and $a(\neq 0)$, $b$, $c$ are constants. In addition, by employing Nevanlinna theory, we show that $g$ must be linear when $f$ is transcendental and $a(\neq 0)$, $b$, $c$ are polynomials in $\mathbb {C}$.