Carath\'eodory extremal functions on the symmetrized bidisc

TL;DR

This paper uses realization theory to solve the Carathéodory extremal problem on the symmetrized bidisc, providing uniqueness results and explicit formulas for extremal functions in various cases.

Contribution

It introduces a realization-theoretic approach to find solutions and formulas for the Carathéodory extremal problem on the symmetrized bidisc, including non-generic cases.

Findings

Solutions are generically unique up to disc automorphisms.

Explicit formulas for extremal functions for non-generic tangent problems.

The approach simplifies finding extremal functions on the symmetrized bidisc.

Abstract

We show how realization theory can be used to find the solutions of the Carath\'eodory extremal problem on the symmetrized bidisc \[ G \stackrel{\rm{def}}{=} \{(z+w,zw):|z|<1, \, |w|<1\}. \] We show that, generically, solutions are unique up to composition with automorphisms of the disc. We also obtain formulae for large classes of extremal functions for the Carath\'eodory problems for tangents of non-generic types.