Spectral Nevanlinna-Pick problem and weak extremals in the symmetrized bidisc

TL;DR

This paper investigates extremal mappings in the symmetrized bidisc, proving they are rational and inner, thereby resolving a previously posed question and advancing understanding of complex geometric function theory.

Contribution

It demonstrates that all m-extremal mappings in the symmetrized bidisc are rational and inner, answering an open question in the field.

Findings

m-extremal mappings are rational functions

m-extremal mappings are -inner functions

The results resolve a question from prior research

Abstract

The main goal of the paper is to study $m$-extremal mappings in the symmetrized bidisc showing that they are rational and $\GG_2$-inner which, in particular, answers a question posed in \cite{Agl-Lyk-You 2013}.