On Linear Combinations of Harmonic Mappings Convex in the Horizontal Direction
Abstract
The process of creating univalent harmonic mappings which are not analytic is not simple or straightforward. One efficient method for constructing desired univalent harmonic maps is by taking the linear combination of two suitable harmonic maps. In this study, we take into account two harmonic, univalent, and convex in the horizontal direction mappings, which are horizontal shears of $\Psi_{m}(z)=\frac{1}{2i\sin \gamma_{m}}\log \left( \frac{ 1+ze^{i\gamma_{m}}}{%
1+ze^{-^{i\gamma_{m}}}}\right),$ and have dilatations $\omega _{1}(z)=z,$
$\omega _{2}(z)=\frac{z+b}{1+bz},$ $b\in (-1,1).$ We obtain sufficient conditions for the linear combination of these two harmonic mappings to be univalent and convex in the
horizontal direction. In addition, we provide an example to illustrate the
result graphically with the help of Maple.
Keywords
Convex in one direction, Harmonic, Linear combination, Univalent
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