On a functional equation related to an automorphism of a unit circle

Abstract

In this article the complete description of the decisions of a functional equation
(1 + ¯ζe^iθ z)^pf (z) = f (ω(0))f (z)
is given, where ω(z) = (e^iθ z + ζ )/(1 + ¯ζe^iθ z) – automorphism of a unit circle E and the decisions are searched among analytical in E functions. It is established, that research of a given functional equation is closely connected to property of stationary points of automorpism ω(z).