Convolution properties of a class of starlike functions

Abstract:

Let $R$ denote the class of functions $f(z) = z + {a_2}{z^2} + \cdots$ that are analytic in the unit disc $E = \{ z:\left | z \right | < 1\}$ and satisfy the condition $\operatorname {Re} (f’(z) + zf''(z)) > 0,z \in E$. It is known that $R$ is a subclass of ${S_t}$, the class of univalent starlike functions in $E$. In the present paper, among other things, we prove (i) for every $n \geq 1$, the $n$th partial sum of $f \in R,{s_n}(z,f)$, is univalent in $E$, (ii) $R$ is closed with respect to Hadamard convolution, and (iii) the Hadamard convolution of any two members of $R$ is a convex function in $E$.