Coman conjecture for the bidisc

TL;DR

This paper proves that the Lempert function equals the Green function with two equal-weight poles in the bidisc, confirming Coman's conjecture in this fundamental case and connecting it to the Nevanlinna-Pick problem.

Contribution

It establishes the equality between the Lempert and Green functions with two poles of equal weights in the bidisc, solving a key case of Coman's conjecture.

Findings

Lempert function equals Green function with two equal poles in the bidisc

Proves a specific case of Coman's conjecture

Links the result to the Nevanlinna-Pick problem in the bidisc

Abstract

In the paper we show the equality between the Lempert function and the Green function with two poles with equal weights in the bidisc thus giving the positive answer to a conjecture of Coman in the simplest unknown case. Actually, a slightly more general equality is proven which in some sense is natural when studied from the point of view of the Nevanlinna-Pick problem in the bidisc.