Univalent Harmonic Mappings Into Two-Slit Domains

Abstract

We study some classes of planar harmonic mappings produced with the shear construction devised by Clunie and Sheil-Small in 1984. The first section reviews the basic concepts and describes the shear construction. The main body of the paper deals with the geometry of the classes constructed.2000 Mathematics subject classification: primary 30C45.

Harmonic shears of slit and polygonal mappings

“…Theorem 2.1. Let ϕ be given by (6). Then the harmonic shear f = h + g, where h is given in Table 1 and Table 2 and the anti-analytic part g = h − ϕ, maps the unit disk D univalently onto a domain which is convex in the horizontal direction.…”

“…which maps D onto C minus four symmetric half-lines. In [6], Ganczar and Widomski have studied some special cases of this mapping and its harmonic shears. Analytic examples of harmonic shears of ϕ with dilatations…”

On convex combinations of harmonic mappings

“…where ω f k is the dilatation of f k . Thus, it is seen that f k satisfy (2.14) and ( The mapping Φ 0,ν maps D onto a domain with parallel slits along the real direction and its harmonic shears along the real direction were studied in [6]. In the next result we find sufficient conditions for the directional convexity of the convex combination of harmonic shears of Φ μ,ν .…”

A Generalisation of the Clunie–sheil-Small Theorem

“…This theorem turns out to be a useful tool both as a univalence criterion and as a method of constructing harmonic mappings. In particular, it plays an important role in the study of harmonic mappings onto polygonal domains [5,10,15], onto a horizontal strip [9], onto a plane with a slit [12] and onto a plane with several slits [3,6,8]. Further interesting examples of harmonic mappings obtained in this way can be also found in [4,7,16].…”