Some results of the class functions with bounded radious rotation

Abstract

Let A be the family of functions f(z) = z + a2z 2 + ... which are analytic in the open unit disc D = {z : |z| < 1}, and denote by P of functions p(z) = z + p1z + p2z 2 + ... analytic in D such that p(z) is in P if and only if p(z) ≺ 1 + z 1 − z ⇔ p(z) = 1 + φ(z) 1 − φ(z) , for some Schwarz function φ(z) and every z ∈ D. Let f(z) be an element of A, and satisfies the condition z f ′ (z) f(z) =

k 4 + 1 2

p1(z) −

k 4 − 1 2

p2(z) where p1(z), p2(z) ∈ P and k ≥ 2, then f(z) is called function with bounded radius rotation. The class of such functions is denoted by Rk. This class is generalization of starlike functions. The main purpose is to give some properties of the class Rk.