International Journal of Differential Equations
Volume 2009 (2009), Article ID 597292, 18 pages
Research Article

An Extension to the Owa-Srivastava Fractional Operator with Applications to Parabolic Starlike and Uniformly Convex Functions

School of Mathematical Sciences, Faculty of Science and Technology, University Kebangsaan Malaysia, Bangi 43600, Selangor, Malaysia

Received 8 October 2008; Accepted 6 January 2009

Academic Editor: Shaher Momani

Copyright © 2009 Oqlah Al-Refai and Maslina Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let 𝒜 be the class of analytic functions in the open unit disk 𝕌. We define Θα,β:𝒜𝒜 by (Θα,βf)(z):=Γ(2α)zαDzα(Γ(2β)zβDzβf(z)),(α,β2,3,4), where Dzγf is the fractional derivative of f of order γ. If α,β[0,1], then a function f in 𝒜 is said to be in the class SPα,β if Θα,βf is a parabolic starlike function. In this paper, several properties and characteristics of the class SPα,β are investigated. These include subordination, characterization and inclusions, growth theorems, distortion theorems, and class-preserving operators. Furthermore, sandwich theorem related to the fractional derivative is proved.