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Published:
Oct 29, 2026DOI:
10.4314/mejs.v18i2.9Keywords:
Article Details
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Main Article Content
Univalent Harmonic Maps on the Unit Disk with Singular Inner and Infinite Blaschke Product Dilatations
Gebreslassie Atsbha Weldegebrial
Hunduma Legesse Geleta
Abstract
We show that a univalent harmonic mapping exhibits a jump when its dilatation is an infinite Blaschke product and it maps the open unit disk univalently into the interior of a convex curve. The results are obtained using boundary behavior analysis of harmonic mappings together with properties of inner functions and their dilatations. We identify conditions on the boundary values of the harmonic function under which the dilatation is an infinite Blaschke product and a singular inner function. These characterizations clarify the relationship between boundary behavior and the analytic structure of the dilatation in univalent harmonic mappings. Examples are provided and visualized using Mathematica, and the examples demonstrate that the converses of the main results do not hold in general.
