Convolution properties of some classes of analytic functions

S. Ponnusamy ; Vikramaditya Singh

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 13, Tome 226 (1996), pp. 138-154

Résumé

Let $\mathcal A$ denote the class of functions analytic in $|z|1$ normalized so that $f(0)=0$ and $f'(0)=1$ and let $\mathcal R(\alpha,\beta)\subset\mathcal A$ be the class of functions $f$ such that $$ \operatorname{Re}[f'(z)+\alpha zF''(z)]>\beta,\qquad\operatorname{Re}\alpha>0,\quad\beta1. $$ We determine conditions so that (i) $f\in\mathcal R(\alpha_1,\beta_1)$, $g\in\mathcal R(\alpha_2,\beta_2)$ implies $f*g$, convolution of $f$ and $g$, is convex; (ii) $f\in\mathcal R(0,\beta_1)$, $g\in\mathcal R(0,\beta_2)$ implies $f*g$ is starlike; (iii) $f\in\mathcal A$ satisfying $f'(z)[f(z)/z]^{\mu-1}\prec1+\lambda z$, $\mu>0$, $0\lambda1$ is starlike and (iv) $f\in\mathcal A$ satisfying $f'(z)+\alpha zf''(z)\prec1+\delta z$, $\alpha>0$, $\delta>0$ is convex or starlike. Bibl. 16 titles.