On classes of analytic functions related to conic domains

We introduce classes of analytic functions related to conic domains, using a new linear multiplier fractional differential operator D n,α λ (n ∈ N0 = {0, 1, . . .}, 0 α <1, λ 0), which is defined as D0f (z) = f (z), D 1,α λ f (z) = (1−λ)Ωαf (z)+λz Ωαf (z) = Dα λ f (z) , D 2,α λ f (z) = Dα λ D 1,α λ f (z) , ... D n,α λ f (z) = Dα λ D n−1,α λ f (z) , where Ωαf (z) = (2−α)zαDα z f (z), and Dα z is the known fractional derivative. Basic properties of these classes are studied, such as inclusion relations and coefficient bounds. Various known or new special cases of our results are also pointed out. © 2007 Elsevier Inc. All rights reserved