Functions with image in a strip
Tirthankar Bhattacharyya, Anthony G. O'Farrell, Shubham Rastogi and, Vijaya Kumar U

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.
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