On rationally loxodromic holomorphic functions

Abstract

We consider a functional equation of the form $f(qz) = R(z)f(z)$, where $R(z)$ is a rational function, $z \in C\setminus \{ 0\},\;q \in C\setminus \{ 0\},\; | q| < 1$. Holomorphic solutions of this equation are obtained. These solutions can be regarded as generalizations of p-loxodromic functions.