Characterizations of Some Domains via Carath\xe9odory Extremals

Abstract

In this paper we characterize the unit disc, the bidisc and the symmetrized bidisc G={(z+w,zw):|z|<1,|w|<1}\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} G =\\{(z+w,zw):|z|<1,\\ |w|<1\\} \\end{aligned}$$\\end{document}in terms of the possession of small classes of analytic maps into the unit disc that suffice to solve all Carathéodory extremal problems in the domain.