Certain subclass of starlike functions

Let R(β, α) denote the class of functions of the form:f(z)=z+a2z2+a3z3+⋯,f(z)=z+a2z2+a3z3+⋯,which are analytic in the open unit disk D = {z : ∣z∣ < 1} and satisfy the conditionRe{f′(z)+αzf″(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.