New Results for a Class of Univalent Functions

Let A denote the family of all analytic functions f ( z ) in the unit disk D = { z ∈ ℂ : ∣ z ∣ < 1} normalized by the conditions f (0) = 0 and f′ (0) = 1. Let U denote the set of all functions f ∈ A satisfying the condition | ( z f ( z ) ) 2 f ′ ( z ) − 1 | < 1 for z ∈ D . Let Ω be the class of all f ∈ A for which | z f ′ ( z ) − f ( z ) | < 1 2 , z ∈ D . In this paper, the relations between the two classes are discussed. Furthermore, some new results on the class Ω are obtained.