Some holomorphic generalizations of loxodromic functions

Abstract

The functional equation of the form $f(qz) = p(z)f(z), z \in C\setminus \{ 0\} , q \in C\setminus \{ 0\} , | q| < 1$ is considered. For certain fixed elementary functions $p(z)$, holomorphic solutions of this equation are found. These solutions are some generalizations of loxodromic functions. Some of solutions are represented via the Schottky – Klein prime function.