Coman conjecture for the bidisc

Łukasz Kosiński, Pascal J. Thomas and Włodzimierz Zwonek

Vol. 287 (2017), No. 2, 411–422

DOI: 10.2140/pjm.2017.287.411

Abstract

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, we prove a slightly more general equality which in some sense is natural when studied from the point of view of the Nevanlinna–Pick problem in the bidisc.

Keywords

Green function, Lempert function, Carathéodory pseudodistance, Coman conjecture, m-extremal, m-complex geodesic, bidisc

Mathematical Subject Classification 2010

Primary: 32U35

Secondary: 30E05

Milestones

Received: 16 November 2014

Revised: 21 July 2015

Accepted: 22 May 2016

Published: 9 March 2017