Injectivity of sections of convex harmonic mappings and convolution theorems

Li, Liulan ; Ponnusamy, Saminathan

Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 331-350.

Résumé

We 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 $n\geq 2$ 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 it is convex in a smaller disk.

Détail

DOI : 10.1007/s10587-016-0259-9

Classification : 30C45
Mots-clés : harmonic mapping; partial sum; univalent mapping; convex mapping; starlike mapping; close-to-convex mapping; harmonic convolution; direction convexity preserving map