Starlikeness and convexity of a class of analytic functions

Let be the class of analytic functions in the unit disk that are normalized with f (0)= f ′(0)− 1 = 0 and let −1 ≤ B < A ≤ 1. In this paper we study the class Gλ,α = { f ∈ : |(1− α + αz f (z)/ f ′(z))/z f ′(z)/ f (z)− (1− α)| < λ, z ∈ }, 0 ≤ α ≤ 1, and give sharp sufficient conditions that embed it into the classes S∗[A,B] = { f ∈ : z f ′(z)/ f (z) ≺ (1 +Az)/(1 + Bz)} and K(δ) = { f ∈ : 1 + z f (z)/ f ′(z) ≺ (1− δ)(1 + z)/(1− z) + δ}, where “≺ ” denotes the usual subordination. Also, sharp upper bound of |a2| and of the Fekete-Szegö functional |a3−μa2| is given for the class Gλ,α.