On the continuation of locally operator monotone functions

TL;DR

This paper extends the concept of continuation from complex analysis to locally operator monotone functions, establishing that such continuations depend solely on domain geometry and generalizing key inequalities for this class.

Contribution

It introduces a geometric-based continuation framework for locally operator monotone functions and generalizes the Julia inequality for Pick functions.

Findings

Continuations depend only on domain geometry.

Established a generalized Julia inequality.

Extended complex analysis concepts to operator monotone functions.

Abstract

We generalize the phenomenon of continuation from complex anal- ysis to locally operator monotone functions. Along the lines of the egde-of- the-wedge theorem, we prove continuations exist dependent only on geometric features of the domain and, namely, independent of the function values. We prove a generalization of the Julia inequality for a class of functions containing locally operator monotone functions, Pick functions.