Completely monotone sequences and harmonic mappings

In the present paper, we will study geometric properties of harmonic mappings whose analytic and co-analytic parts are (shifted) generated functions of completely monotone sequences.

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.

A Schwarz lemma for hyperbolic harmonic mappings in the unit ball

Assume that $p\in [1,\infty ]$ and $u=P_{h}[\phi ]$, where $\phi \in L^{p}(\mathbb{S}^{n-1},\mathbb{R}^n)$ and $u(0) = 0$. Then we obtain the sharp inequality $\lvert u(x) \rvert \le G_p(\lvert x \rvert )\lVert \phi \rVert_{L^{p}}$ for some smooth function $G_p$ vanishing at $0$. Moreover, we obtain an explicit form of the sharp constant $C_p$ in the inequality $\lVert Du(0)\rVert \le C_p\lVert \phi \rVert \le C_p\lVert \phi \rVert_{L^{p}}$. These two results generalize and extend some known results from the harmonic mapping theory (D. Kalaj, Complex Anal. Oper. Theory 12 (2018), 545–554, Theorem 2.1) and the hyperbolic harmonic theory (B. Burgeth, Manuscripta Math. 77 (1992), 283–291, Theorem 1).

Convolutions of Harmonic Mappings Convex in the Horizontal Direction

In this paper, we establish some results concerning the convolutions of harmonic mappings convex in the horizontal direction with harmonic vertical strip mappings. Furthermore, we provide examples illustrated graphically with the help of Maple to illuminate the results.