Let Ω denote the class of functions w(z), w(0) = 0, |w(z)| < 1 analytic in the unit disc ⋃ = {z : |z| < 1}. For arbitrary fixed numbers A, B, −1 < A ≤ 1, −1 ≤ B < 1 and 0 ≤ α < p, denote by P(A, B, p, α) the class of functions analytic in ⋃ such that P(z) ϵ P(A, B, p, α) if and only if , w ϵ Ω, z ϵ ⋃. Moreover, let S(A, B, p, α) denote the class of functions analytic in ⋃ and satisfying the condition that f(z) ϵ S(A, B, p, α) if and only if for some P(z) ϵ P(A, B, p, α) and all z in ⋃.In this paper we determine the bounds for |f(z)| and in S(A, B, p, α), we investigate the coefficient estimates for functions of the class S(A, B, p, α) and we study some properties of the class S(A, B, p, α).