Convolution Properties of a Subclass of Univalent Harmonic Mappings

Abstract

Sudharshan et al. in \cite{Sud} studied a subclass $TS_{H}^{*}(m,n,\lambda,\gamma)$ of univalent harmonic mappings of the form $f_{m}=h+\overline{g_{m}}$, where $$ h(z)=z-\sum\limits_{k=2}^{\infty} |a_{k}| z^{k}, \;\;\ g_{m}(z) = (-1)^{m-1} \sum\limits_{k=1}^{\infty} |b_{k}| \overline{z}^{k}$$ using Salagean modified differential operator. They proved that for $0\leq \gamma_{1} \leq \gamma_{2} <1,$ if $f_{m} \in TS_{H}^{*}(m,n,\lambda,\gamma_{1})$ and $F_{m} \in TS_{H}^{*}(m,n,\lambda,\gamma_{2})$, then $f_{m} \ast F_{m} \in TS_{H}^{*}(m,n,\lambda,\gamma_{1})$. In the present article, we improved the above stated result.