Applications of fractional calculus to parabolic starlike and uniformly convex functions

Let A be the class of analytic functions in the open unit disk U. Given 0 ≤ λ < 1, let Ωλ be the operator defined on A by (ωλf)(z)=Γ(2-λ)zλDλzf(z), where Dλzf is the fractional derivative of f of order λ. A function f in A is said to be in the class SPλ if Ωλf is a parabolic starlike function. In this paper, several basic properties and characteristics of the class SPλ are investigated. These include subordination, inclusion, and growth theorems, class-preserving operators, Fekete-Szeg problems, and sharp estimates for the first few coefficients of the inverse function.