Functions with image in a strip

TL;DR

This paper studies holomorphic functions mapping the unit disc into a strip in the complex plane, establishing conditions under which such functions are constant, and explores operator-valued versions with applications to semigroup theory.

Contribution

It introduces conditions for holomorphic functions into a strip to be constant and extends these results to operator-valued functions with applications to semigroups.

Findings

Functions into a strip are constant under certain conditions

Operator-valued versions are developed

Applications to semigroup theory are provided

Abstract

We consider holomorphic functions on the unit disc whose images are contained in a strip of the complex plane. Under an additional condition, such functions are constants. We also consider appropriate operator valued versions. Applications are found to the theory of semigroups.