International Journal of Differential Equations

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

Abstract

𝒜 Let $\mathscr{A}$ be the class of analytic functions in the open unit disk . We define ${\Theta }^{\alpha ,\beta }:\mathscr{A}\to \mathscr{A}$ by $({\Theta }^{\alpha ,\beta }f)$$(z):=\Gamma (2-\alpha ){z}^{\alpha }{D}_{z}^{\alpha }$$(\Gamma (2-\beta ){z}^{\beta }{D}_{z}^{\beta }f(z)),$$(\alpha ,\beta \ne 2,3,4\dots ),$ where ${D}_{z}^{\gamma }f$ is the fractional derivative of $f$ of order $\gamma$. If $\alpha ,\beta \in [0,1]$, then a function $f$ in $\mathscr{A}$ is said to be in the class $\text{S}{\text{P}}_{\alpha ,\beta }$ if ${\Theta }^{\alpha ,\beta }f$ is a parabolic starlike function. In this paper, several properties and characteristics of the class $\text{S}{\text{P}}_{\alpha ,\beta }$ 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.

Article information

Source
Int. J. Differ. Equ., Volume 2009 (2009), Article ID 597292, 18 pages.

Dates
Accepted: 6 January 2009
First available in Project Euclid: 26 January 2017

https://projecteuclid.org/euclid.ijde/1485399811

Digital Object Identifier
doi:10.1155/2009/597292

Mathematical Reviews number (MathSciNet)
MR2525713

Zentralblatt MATH identifier
1202.26012

Citation

Al-Refai, Oqlah; Darus, Maslina. An Extension to the Owa-Srivastava Fractional Operator with Applications to Parabolic Starlike and Uniformly Convex Functions. Int. J. Differ. Equ. 2009 (2009), Article ID 597292, 18 pages. doi:10.1155/2009/597292. https://projecteuclid.org/euclid.ijde/1485399811

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