An Extension to the Owa-Srivastava Fractional Operator with Applications to Parabolic Starlike and Uniformly Convex Functions
Let 𝒜 be the class of analytic functions in the open unit disk . We define Θα,β:𝒜→𝒜 by (Θα,βf)(z):=Γ(2−α)zαDzα(Γ(2−β)zβDzβf(z)),(α,β≠2,3,4…), 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...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2009-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2009/597292 |
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| Summary: | Let 𝒜 be the class of analytic functions in the open unit disk . We define Θα,β:𝒜→𝒜 by (Θα,βf)(z):=Γ(2−α)zαDzα(Γ(2−β)zβDzβf(z)),(α,β≠2,3,4…), 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. |
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| ISSN: | 1687-9643 1687-9651 |