International Journal of Differential Equations

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

Oqlah Al-Refai and Maslina Darus

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Abstract

𝒜 Let 𝒜 be the class of analytic functions in the open unit disk . We define Θ α , β : 𝒜 𝒜 by ( Θ α , β f ) ( z ) : = Γ ( 2 α ) z α D z α ( Γ ( 2 β ) z β D z β f ( z ) ) , ( α , β 2 , 3 , 4 ) , where D z γ f 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 S P α , β if Θ α , β f is a parabolic starlike function. In this paper, several properties and characteristics of the class S P α , 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.

Article information

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

Dates
Received: 8 October 2008
Accepted: 6 January 2009
First available in Project Euclid: 26 January 2017

Permanent link to this document
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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References

  • A. W. Goodman, “On uniformly convex functions,” Annales Polonici Mathematici, vol. 56, no. 1, pp. 87–92, 1991.MR1145573
  • W. C. Ma and D. Minda, “Uniformly convex functions,” Annales Polonici Mathematici, vol. 57, no. 2, pp. 165–175, 1992.MR1182182
  • F. Rønning, “Uniformly convex functions and a corresponding class of starlike functions,” Proceedings of the American Mathematical Society, vol. 118, no. 1, pp. 189–196, 1993.MR1128729
  • B. C. Carlson and D. B. Shaffer, “Starlike and prestarlike hypergeometric functions,” SIAM Journal on Mathematical Analysis, vol. 15, no. 4, pp. 737–745, 1984.MR747433
  • S. Owa and H. M. Srivastava, “Univalent and starlike generalized hypergeometric functions,” Canadian Journal of Mathematics, vol. 39, no. 5, pp. 1057–1077, 1987.MR918587
  • S. Owa, “On the distortion theorems–-I,” Kyungpook Mathematical Journal, vol. 18, no. 1, pp. 53–59, 1978.MR0507718
  • H. M. Srivastava and S. Owa, “An application of the fractional derivative,” Mathematica Japonica, vol. 29, no. 3, pp. 383–389, 1984.MR752235
  • H. M. Srivastava and S. Owa, Eds., Univalent Functions, Fractional Calculus, and Their Applications, Ellis Horwood Series: Mathematics and Its Applications, Halsted Press/John Wiley & Sons, New York, NY, USA, 1989.MR1199135
  • H. M. Srivastava and A. K. Mishra, “Applications of fractional calculus to parabolic starlike and uniformly convex functions,” Computers & Mathematics with Applications, vol. 39, no. 3-4, pp. 57–69, 2000.MR1740907
  • S. S. Miller and P. T. Mocanu, “Subordinants of differential superordinations,” Complex Variables: Theory and Application, vol. 48, no. 10, pp. 815–826, 2003.MR2014390
  • O. Al-Refai and M. Darus, “Main differential sandwich theorem with čommentComment on ref. [11?]: Please update the information of this reference, if possible. some applications,” Lobachevskii Journal of Mathematics, in press, 2009.
  • St. Ruscheweyh and T. Sheil-Small, “Hadamard products of Schlicht functions and the Pólya-Schoenberg conjecture,” Commentarii Mathematici Helvetici, vol. 48, no. 1, pp. 119–135, 1973.MR0328051
  • St. Ruscheweyh and J. Stankiewicz, “Subordination under convex univalent functions,” Bulletin of the Polish Academy of Sciences, Mathematics, vol. 33, no. 9-10, pp. 499–502, 1985.MR826376
  • Y. Ling and S. Ding, “A class of analytic functions defined by fractional derivation,” Journal of Mathematical Analysis and Applications, vol. 186, no. 2, pp. 504–513, 1994.MR1293007
  • G. M. Goluzin, “On the majorization principle in function theory,” Doklady Akademii Nauk SSSR, vol. 42, pp. 647–650, 1935 (Russian).
  • Ch. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, Germany, 1975.MR0507768
  • S. D. Bernardi, “Convex and starlike univalent functions,” Transactions of the American Mathematical Society, vol. 135, pp. 429–446, 1969.MR0232920