A characterization of M\"obius transformations

TL;DR

This paper characterizes Möbius transformations among inner functions by showing their derivatives are outer functions, providing a new criterion involving a reverse Schwarz--Pick estimate.

Contribution

It offers a novel characterization of Möbius transformations using the outer property of derivatives and a reverse Schwarz--Pick estimate.

Findings

Derivative of an inner function is outer iff the function is a Möbius transformation.

Provides an alternative characterization via a reverse Schwarz--Pick estimate.

Establishes a new criterion for identifying Möbius transformations.

Abstract

We prove that the derivative $\theta'$ of an inner function $\theta$ is outer if and only if $\theta$ is a M\"obius transformation. An alternative characterization involving a reverse Schwarz--Pick type estimate is also given.