Complex Geodesics in Tube Domains and Their Role in the Study of Harmonic Mappings in the Disc

Abstract

We continue the research on the structure of complex geodesics in tube domains over (bounded) convex bases. In some special cases a more explicit form of the geodesics than the existing ones are provided. As one of the consequences of our study an effective formula for the Kobayashi—Royden metric in the tube domain \({T_{{\mathbb{B}_n}}}\) at the origin is given. The results on the Kobayashi—Royden metric in a natural way provide versions of the Schwarz Lemma for harmonic mappings. We also present a result on harmonic mappings defined on the disc that may be seen as a generalisation of the Radó—Kneser—Choquet Theorem for a class of harmonic bivalent mappings that lets understand better the geometry of complex geodesics in tube domains.