The boundary behaviour of harmonic univalent maps

Yusuf Abu-Muhanna and Abdallah Khalil Lyzzaik

Vol. 141 (1990), No. 1, 1–20

DOI: 10.2140/pjm.1990.141.1

Abstract

Let D denote the open unit disc in the complex plane and f = h + ḡ a complex-valued, harmonic, univalent and orientation preserving map in D, where h and g are analytic in D. We show that g,h ∈ H^λ and f ∈ h^λ for some λ > 0, where H^λ (h^λ) is the Hardy space of order λ for analytic (harmonic) functions. We also study the correspondence under f between ∂D (boundary of D) and the prime ends of f(D).

Mathematical Subject Classification 2000

Primary: 30D40

Secondary: 30D55, 31A20

Milestones

Received: 7 January 1988

Revised: 11 October 1988

Published: 1 January 1990