Convolution theorems for starlike and convex functions in the unit disc
Volume 84 / 2004
Abstract
Let ${\cal A}$ denote the space of all analytic functions in the unit disc ${\mit\Delta} $ with the normalization $f(0)=f'(0)-1=0$. For $\beta <1$, let $${\cal P}_{\beta}^0=\{f\in {\cal A}: \mathop{\rm Re}\nolimits f'(z)> \beta, \,z\in{\mit\Delta}\}.$$ For $\lambda > 0$, suppose that $\cal F$ denotes any one of the following classes of functions: $$\eqalign{M_{1,\lambda}^{(1)}&=\{f\in {\cal A}:\mathop{\rm Re}\nolimits\{ z(zf'(z))' '\}> -\lambda , \, z\in {\mit\Delta} \},\cr M_{1,\lambda}^{(2)}&=\{f\in {\cal A}:\mathop{\rm Re}\nolimits\{ z(z^2f' '(z))' '\}> -\lambda , \, z\in {\mit\Delta}\},\cr M_{1,\lambda}^{(3)}&=\{f\in {\cal A}: \mathop{\rm Re}\nolimits \{\textstyle\frac{1}{2}(z(z^2f'(z))' ')'-1 \}> -\lambda, \,z \in {\mit\Delta} \}.\cr}$$ The main purpose of this paper is to find conditions on $\lambda$ and $\gamma$ so that each $f \in {\cal F}$ is in ${\cal S}_\gamma $ or ${\cal K}_\gamma $, $\gamma \in [0,1/2]$. Here ${\cal S}_\gamma $ and ${\cal K}_\gamma $ respectively denote the class of all starlike functions of order $\gamma$ and the class of all convex functions of order $\gamma$. As a consequence, we obtain a number of convolution theorems, namely the inclusions $M_{1,\alpha}*{\cal G} \subset {\cal S}_{\gamma }$ and $M_{1,\alpha}*{\cal G} \subset {\cal K}_{\gamma }$, where $\cal G$ is either ${\cal P}_{\beta}^0$ or $M_{1,\beta}$. Here $M_{1,\lambda}$ denotes the class of all functions $f$ in ${\cal A}$ such that $\mathop{\rm Re}\nolimits(zf' '(z))> -\lambda$ for $z\in{\mit\Delta}$.
