Starlikeness criteria for a certain class of analytic functions

We denote by A , the class of all analytic functions f in the unit disc Δ = { z ∈ C : | z | < 1 } with the normalization f ( 0 ) = f ′ ( 0 ) − 1 = 0 . For a positive number λ > 0 , we denote by U 3 ( λ ) the class of all f ( z ) = z + ∑ n = 2 ∞ a n z n ∈ A , such that a 3 − a 2 2 = 0 , and satisfying the condition | ( z f ( z ) ) 2 f ′ ( z ) − 1 | < λ , z ∈ Δ . A function f ∈ A is said to be in SR ( γ ) if | arg f ′ ( z ) | < π γ / 2 . In this paper, we find conditions on λ , α and γ such that U 3 ( λ ) is included in the class of all starlike functions of order α , or the class of all strongly starlike functions of order γ , or SR ( γ ) , respectively.