Injectivity of sections of convex harmonic mappings and convolution theorems

TL;DR

This paper investigates the injectivity and convolution properties of convex harmonic mappings, establishing conditions under which harmonic convolutions and partial sums are univalent, close-to-convex, or convex within certain disks.

Contribution

It introduces new classes of harmonic functions and proves their convolution preserves univalence and close-to-convexity, also analyzing the convexity of harmonic sections in specific disks.

Findings

Harmonic convolution of certain classes remains univalent and close-to-convex under parameter conditions.

Partial sums of univalent harmonic convex functions are univalent and close-to-convex in |z|<1/4 for n≥2.

Section s_{3,3}(f) is not convex in |z|<1/4 but is convex in a smaller disk.

Abstract

In the article the authors consider the class ${\mathcal H}_0$ of sense-preserving harmonic functions $f=h+\overline{g}$ defined in the unit disk $|z|<1$ and normalized so that $h(0)=0=h'(0)-1$ and $g(0)=0=g'(0)$, where $h$ and $g$ are analytic in the unit disk. In the first part of the article we present two classes $\mathcal{P}_H^0(\alpha)$ and $\mathcal{G}_H^0(\beta)$ of functions from ${\mathcal H}_0$ and show that if $f\in \mathcal{P}_H^0(\alpha)$ and $F\in\mathcal{G}_H^0(\beta)$, then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters $\alpha$ and $\beta$ are satisfied. In the second part we study the harmonic sections (partial sums) $$ s_{n, n}(f)(z)=s_n(h)(z)+\overline{s_n(g)(z)}, $$ where $f=h+\overline{g}\in {\mathcal H}_0$, $s_n(h)$ and $s_n(g)$ denote the $n$-th partial sums of $h$ and $g$, respectively. We prove, among others, that if $f=h+\overline{g}\in{\mathcal H}_0$ is a univalent harmonic convex mapping, then $s_{n, n}(f)$ is univalent and close-to-convex in the disk $|z|< 1/4$ for $n\geq 2$, and $s_{n, n}(f)$ is also convex in the disk $|z|< 1/4$ for $n\geq2$ and $n\neq 3$. Moreover, we show that the section $s_{3,3}(f)$ of $f\in {\mathcal C}_H^0$ is not convex in the disk $|z|<1/4$ but is shown to be convex in a smaller disk.