Certain subclass of starlike functions

Abstract

Let R( β, α) denote the class of functions of the form: f ( z ) = z + a 2 z 2 + a 3 z 3 + ⋯ , which are analytic in the open unit disk D = { z : ∣ z∣ < 1} and satisfy the condition Re { f ′ ( z ) + α zf ″ ( z ) } > β ( α > 0 ; β < 1 ; z ∈ D ) . We find extreme points of R( β, α) and obtain some sharp bounds for certain linear problems. And we find number β( α) ( α ⩾ 1) such that R( β, α) is a subclass of S ∗, which denotes the class consisting of univalent starlike functions in D.