Injectivity of sections of convex harmonic mappings and convolution theorems

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 $h(0)=0=h^{\prime }(0)-1$ and $g(0)=0=g^{\prime }(0)$, 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 $n\ge 2$, and $s_{n,n}(f)$ is also convex in the disk $|z|<1/4$ for $n\ge 2$ and $n\ne 3$. 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.

16 pages, 3 figures; To appear in Czechoslovak Mathematical Journal

References in corpus (1)

• Uniform close-to-convexity radius of sections of functions in the close-to-convex family