Uniqueness of harmonic mappings with Blaschke dilatations

Abstract

Abstract. Let Ω be a bounded convex domain and let ω be a finite Blaschke product of order N = 1, 2, · · · . It is known that the elliptic differential equation f z /f z = ω admits a one-to-one solution normalized by f (0) = 0, f z (0) > 0 and maps the open unit disc D onto a convex (n + 2)−gon whose vertices belong to ∂Ω. In this paper it is shown that this solution is unique.

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“…Harmonic mappings with given dilatation, especially when Beltrami coefficient is Blaschke product, are investigated in many papers. Several results on the existence and the uniqueness of these and more general mappings can be found, for example, in [2], [10] [12], [16].…”

“…If f 1 is harmonic mapping and it satisfies the equation (2.1), by [2] and [16], it is a mapping of the unit disk onto a quadrilateral inscribed in the unit disk, which is a Poisson extension of a step function, determined by a set of four points on the unit circle, that defines a quadrilateral in the domain. In [16] it is proved that the sum of lengths of two non-adjacent sides of quadrilateral in co-domain are equal.…”