Estimates of holomorphic functions in zero-free domains

TL;DR

This paper derives lower bounds for the magnitude of holomorphic functions in the upper half-plane that are zero-free within a specific strip, including scalar and operator-valued cases, with implications for complex analysis and operator theory.

Contribution

It provides new lower bound estimates for zero-free holomorphic functions in a strip, extending to operator-valued functions with trace class properties.

Findings

Established lower bounds for scalar zero-free holomorphic functions.

Extended bounds to operator-valued functions with trace class operators.

Analyzed functions that are unitary on the real line within the strip.

Abstract

We study functions f(z) holomorphic in the upper half plane and having no zeros when the imaginary part of z is between 0 and 1, and we obtain a lower bound for the modulus of f(z) in this strip. In our analysis we deal with scalar functions f(z) as well as with operator valued holomorphic functions I+A(z) assuming that A(z) is a trace class operator in the upper half plane and I+A(z) is invertible in the same strip and is unitary on the real line.