Abstract
In this paper, we study harmonic mappings by using the shear construction, introduced by Clunie and Sheil-Small in 1984. We consider two classes of conformal mappings, each of which maps the unit disk D univalently onto a domain which is convex in the horizontal direction, and shear these mappings with suitable dilatations ω. Mappings of the first class map the unit disk D onto four-slit domains and mappings of the second class take D onto regular n-gons. In addition, we discuss the minimal surfaces associated with such harmonic mappings. Furthermore, illustrations of mappings and associated minimal surfaces are given by using Mathematica.
Original language | English |
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Pages (from-to) | 588-598 |
Number of pages | 11 |
Journal | Applied Mathematics and Computation |
Volume | 233 |
DOIs | |
State | Published - 1 Mar 2014 |
Externally published | Yes |
Keywords
- Convex along real directions
- Convex functions
- Harmonic shear
- Harmonic univalent mappings
- Minimal surfaces
- Polygonal mappings
- Slit mappings