Convolution properties of a class of starlike functions

Let R denote the class of functions f(z) = z + a2z2 +* that are analytic in the unit disc E = {z: Izi K 1 } and satisfy the condition Re(f'(z) + zf"(z)) > 0, z E E. It is known that R is a subclass of St, the class of univalent starlike functions in E. In the present paper, among other things, we prove (i) for every n > 1, the nth partial sum of f E R, s,(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.