Convolution properties of a class of starlike functions

Abstract

Let R R denote the class of functions f ( z ) = z + a 2 z 2 + ⋯ f(z) = z + {a_2}{z^2} + \cdots that are analytic in the unit disc E = { z : | z | > 1 } E = \{ z:\left | z \right | > 1\} and satisfy the condition Re ⁡ ( f ′ ( z ) + z f ( z ) ) > 0 , z ∈ E \operatorname {Re} (f’(z) + zf(z)) > 0,z \in E . It is known that R R is a subclass of S t {S_t} , the class of univalent starlike functions in E E . In the present paper, among other things, we prove (i) for every n ≥ 1 n \geq 1 , the n n th partial sum of f ∈ R , s n ( z , f ) f \in R,{s_n}(z,f) , is univalent in E E , (ii) R R is closed with respect to Hadamard convolution, and (iii) the Hadamard convolution of any two members of R R is a convex function in E E .