Upper bound for the Lempert function of smooth domains

TL;DR

This paper provides an upper bound estimate for the Lempert function in smooth bounded domains in complex space, relating it to the boundary distance.

Contribution

It introduces a new upper estimate for the Lempert function for $C^{1+ ext{epsilon}}$-smooth bounded domains based on boundary distance.

Findings

Established an upper bound for the Lempert function in smooth domains.

Connected the Lempert function estimate to boundary distance.

Extended understanding of complex geometric function theory.

Abstract

An upper estimate for the Lempert function of any $C^{1+\epsilon}$-smooth bounded domain in $\Bbb C^n$ is found in terms of the boundary distance.