The Real Schwarz Lemma: The Sequel

TL;DR

This paper explores the real analogue of the Schwarz lemma, demonstrating that real analytic automorphisms of (-1,1) can have arbitrarily large derivatives at zero, and can be constructed to have any specified derivative value.

Contribution

It introduces new families of real analytic automorphisms of (-1,1) with prescribed derivatives, extending previous results on the real Schwarz lemma.

Findings

Existence of real automorphisms with arbitrarily large derivatives at zero.

Construction of functions with any desired derivative at zero.

Open questions on related problems in real analysis.

Abstract

A decade ago, when teaching complex analysis, the third named author posed the question on whether or not there is an analogue to the Schwarz lemma for real analytic functions. This led to the note [MT], indicating that it is possible to have a real analytic automorphism $f$ of $(-1,1)$ with $f'(0)$ arbitrarily large. In this note we provide other families with this property, and moreover show that we can always find such a function so that $f'(0)$ equals any desired real number. We end with some questions on related problems.