On some alpha-convex functions

Abstract

In this paper, we define a general class of α-convex functions, denoted by MLβ,α(q), with respect to a convex domain D (q(z) ∈ Hu(U), q(0) = 1 , q(U) = D) contained in the right half plane by using the linear operator D β λ defined by D β λ : A → A , D β λ f(z) = z + X∞ j=2 (1 + (j − 1)λ) β ajz j , where β, λ ∈ R, β ≥ 0, λ ≥ 0 and f(z) = z+ X∞ j=2 ajz j . Regarding the class MLβ,α(q), we give a inclusion theorem and a transforming theorem, from which we may obtain many particular results.