Journal The of Geometric Analysis Volume 17, Number 1, 2007
Uniqueness of Harmonic Mappings with Blaschke Dilatations
By D. Bshouty, A. Lyzzaik, and A. Weitsman
ABSTRACT. Let be a bounded convex domain and let be a nite Blaschke product of order N = 1, 2, . It is known that the elliptic differential equation fz /fz = admits a one-to-one solution normalized by f (0) = 0, fz (0) > 0 and maps the open unit disc D onto a convex (n + 2)-gon whose vertices belong to . In this article it is shown that this solution is unique.
1. Introduction
A sense-preserving harmonic mapping f of the open unit disk D is a solution of the elliptic differential equation fz (z) = (z) , fz (z) (1.1)
where , known as the analytic dilatation of f , is an analytic function in D with (D) D. A useful form of f is f =h+g , where h and g are analytic functions in D. In this case, = g / h and the Jacobian of f is J (f ) = h
2
g
2
= h
2
1 is
2
.
It known that f is an open map, and that it is locally one-to-one except possibly at isolated points where it behaves locally like analytic functions near zero derivatives. We call f univalent or locally univalent if it is one-to-one or locally one-to-one in D, respectively. The Riemann mapping theorem (RMT) states: If < k < 1 in D, is a bounded simply connected domain with a locally connected boundary, and if w0 , then there exists a univalent solution f of (1.1) that maps D onto with f (0) = w0 and fz (0) > 0. Moreover, if is a...