Convolution theorems for starlike and convex functions in the unit disc

M. Anbudurai¹ ; R. Parvatham² ; S. Ponnusamy³ ; V. Singh⁴

¹ Department of Mathematics D.G. Vaishnav College Chennai 600 106, India
² The Ramanujan Institute University of Madras Chennai 600 005, India
³ Department of Mathematics Indian Institute of Technology IIT-Madras, Chennai 600 036, India
⁴ 3A/95, Azad Nagar Kanpur 208 002, India

Annales Polonici Mathematici, Tome 84 (2004) no. 1, pp. 27-39.

Résumé

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 $z\in{\mit\Delta}$.

Détail

DOI : 10.4064/ap84-1-2

Keywords: cal denote space analytic functions unit disc mit delta normalization beta cal beta cal mathop nolimits beta mit delta lambda suppose cal denotes following classes functions eqalign lambda cal mathop nolimits lambda mit delta lambda cal mathop nolimits lambda mit delta lambda cal mathop nolimits textstyle frac lambda mit delta main purpose paper conditions lambda gamma each cal cal gamma cal gamma gamma here cal gamma cal gamma respectively denote class starlike functions order gamma class convex functions order gamma consequence obtain number convolution theorems namely inclusions alpha * cal subset cal gamma alpha * cal subset cal gamma where cal either cal beta beta here lambda denotes class functions cal mathop nolimits lambda mit delta