Let Tn(A, B, γ, α) (−1 ≤ B < 1, B < A, 0 < γ ≤ 1 and α > 0) denote the class of functions of the form which are analytic in the open unit disk U and satisfy the following subordination condition f′(z) + αzf′′(z)≺((1 + Az)/(1 + Bz)) γ, for(z ∈ U; A ≤ 1; 0 < γ < 1), (1 + Az)/(1 + Bz), for(z ∈ U; γ = 1). We obtain sharp bounds on Re f′(z), Re f(z)/z, |f(z)|, and coefficient estimates for functions f(z) belonging to the class Tn(A, B, γ, α). Conditions for univalency and starlikeness, convolution properties, and the radius of convexity are also considered.