A Class of Analytic Functions Defined by Fractional Derivation

Let ƒ(z)=z+... be analytic in the unit disc |z| < 1 and 0 ≤ λ ≤ 1. We define a linear operator by Qλƒ=Γ(2−λ)zλDλzƒ(z), where Dλzƒ(z) denotes the fractional derivative of ƒ(z). The function ƒ(z) is said to be in R(λ, α) if it satisfies the condition Re{Qλƒ/z}>α, 0 ≤ α < 1, |z| < 1. In this paper, we prove, for 0 ≤ u ≤ λ < 1, that R(λ, α)⊆R(u, α) and study some subordination properties. We also obtain distortion theorems and a coefficient inequality for R(λ, α). Finally, we discuss the Hadamard product of the class R(λ, α).