Harmonic mappings and its directional convexity

For any $\mu _{j}\ (\mu _{j}\in \mathbb{C},\left\vert \mu _{j}\right\vert =1,j=1,2)$, we consider the rotations $f_{\mu _{1}}$ and $F_{\mu _{2}}$ of right half-plane harmonic mappings $f,F\in S_{\mathcal{H}}$ which are CHD with the prescribed dilatations $\omega _{f}(z)=\left( a-z\right) /\left(1-az\right) $ for some $a$ $\left( -1<a<1\right) $ and $\omega _{F}(z)=$ $e^{i\theta }z^{n}$ $\left( n\in \mathbb{N},\theta \in \mathbb{R}\right) $, $\omega _{F}(z)=$ $\left( b-z\right) /\left( 1-bz\right) $, $\omega_{F}(z)=\left( b-ze^{i\phi }\right) /\left( 1-bze^{i\phi }\right) $ $(-1<b<1,\phi \in \mathbb{R})$, respectively. It is proved that the convolution $f_{\mu _{1}}\ast F_{\mu _{2}}\in S_{\mathcal{H}}$ and is convex in the direction of $\overline{\mu _{1}\mu _{2}}$ under certain conditions on the parameters involved.

Topology of directional convexity

Herein, we have proven a Fink – Wood conjecture that if Oʹ is the closure of some orientation set O, then a set is a directed O-halfspace if and only if it is a directed Oʹ-halfspace.

Directional Convexity of Convolutions of Harmonic Functions

Harmonic functions can be constructed using two analytic functions acting as their analytic and coanalytic parts but the prediction of the behavior of convolution of harmonic functions, unlike the convolution of analytic functions, proved to be challenging. In this paper we use the shear construction of harmonic mappings and introduce dilatation conditions that guarantee the convolution of two harmonic functions to be harmonic and convex in the direction of imaginary axis.