Convolution properties of some classes of analytic functions

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
$$ \operatorname{Re}[f'(z)+\alpha zF''(z)]>\beta,\qquad\operatorname{Re}\alpha>0,\quad\beta<1. $$
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 $f'(z)[f(z)/z]^{\mu-1}\prec1+\lambda z$, $\mu>0$, $0<\lambda<1$ 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.