Sandwich results for higher order fractional derivative operators

M. Zayed; T. Bulboaca

doi:10.15330/ms.49.1.52-66

In this paper we obtain some differential subordinations and superordinations related to a generalized fractional derivative operator for higher order derivatives of multivalent functions. Moreover, we derive some sandwich results under certain assumptions on the parameters involved, and these new results generalize some previously well-known theorems.

1. R.M. Ali, V. Ravichandran, K.M. Hussain, K.G. Subramanian, Differential sandwich theorems for certain analytic functions, Far East J. Math. Sci., 15 (2004), №1, 87-94.

2. S.M. Amsheri, V. Zharkova, Differential subordinations and superordinations for p-valent functions defined by fractional derivative operator, Demonstr. Math., 46 (2013), №3, 505-515.

3. S.M. Amsheri, V. Zharkova, Differential sandwich theorems of p-valent functions associated with a certain fractional derivative operator, Kragujevac J. Math., 35 (2011), №3, 387-398.

4. M.K. Aouf, A.O. Mostafa, H.M. Zayed, Some characterizations of integral operators associated with certain classes of p-valent functions defined by the Srivastava-Saigo-Owa fractional differintegral operator, Complex Anal. Oper. Theory, 10 (2016), №6, 1267-1275.

5. M.K. Aouf, A.O. Mostafa, H.M. Zayed, Subordination and superordination properties of p-valent functions defined by a generalized fractional differintegral operator, Quaest. Math., 39 (2016) №4, 545-560.

6. M.K. Aouf, A.O. Mostafa, H.M. Zayed, On certain subclasses of multivalent functions defined by a generalized fractional differintegral operator, Afr. Mat., 28 (2017), №1-2, 99-107.

7. T. Bulboaca, Classes of first-order differential superordinations, Demonstr. Math., 35 (2002), №2, 287-292.

8. T. Bulboaca, A class of superordination-preserving integral operators, Indag. Math. (N.S.), 13 (2002), №3, 301-311.

9. T. Bulboaca, Differential subordinations and superordinations, new results, House of Scientific Boook Publ., Cluj-Napoca, 2005.

10. G.P. Goyal, J.K. Prajapat, A new class of analytic p-valent functions with negative coefficients and fractional calculus operators, Tamsui Oxf. J. Inf. Math. Sci., 20 (2004), №2, 175-186.

11. S.S. Miller, P.T. Mocanu, Differential subordinations: theory and applications, Series on Monographs and Textbooks in Pure and Appl. Math., V.255, Marcel Dekker, Inc., New York, 2000.

12. S.S. Miller, P.T. Mocanu, Subordinations of differential superordinations, Complex Var., 48 (2003) №10, 815-826.

13. A.O. Mostafa, M.K. Aouf, H.M. Zayed, T. Bulboac.a, Multivalent functions associated with Srivastava- Saigo-Owa fractional differintegral operator. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, https://doi.org/10.1007/s13398-017-0436-1, /to appear/

14. S. Owa, On the distortion theorems I. Kyungpook Math. J., 18 (1978), 53-59.

15. W.C. Royster, On the univalence of a certain integral, Michigan Math. J., 12 (1965), №4, 385-387.

16. T.N. Shanmugam, V. Ravichandran, S. Sivasubramanian, Differential sandwich theorems for some subclasses of analytic functions, Aust. J. Math. Anal. Appl., 3 (2006), №1, 1-11.

	Sandwich results for higher order fractional derivative operators
Author	M. Zayed¹, T. Bulboaca² hanaa_zayed42@yahoo.com¹, bulboaca@math.ubbcluj.ro² 1) Department of Mathematics, Faculty of Science Menofia University, Shebin Elkom, Egypt; 2) Faculty of Mathematics and Computer Science Babes-Bolyai University, Cluj-Napoca, Romania
Abstract	In this paper we obtain some differential subordinations and superordinations related to a generalized fractional derivative operator for higher order derivatives of multivalent functions. Moreover, we derive some sandwich results under certain assumptions on the parameters involved, and these new results generalize some previously well-known theorems.
Keywords	analytic functions; univalent functions; differential subordination and superordination; hypergeometric function; generalized fractional derivative operator
DOI	doi:10.15330/ms.49.1.52-66
Reference	1. R.M. Ali, V. Ravichandran, K.M. Hussain, K.G. Subramanian, Differential sandwich theorems for certain analytic functions, Far East J. Math. Sci., 15 (2004), №1, 87-94. 2. S.M. Amsheri, V. Zharkova, Differential subordinations and superordinations for p-valent functions defined by fractional derivative operator, Demonstr. Math., 46 (2013), №3, 505-515. 3. S.M. Amsheri, V. Zharkova, Differential sandwich theorems of p-valent functions associated with a certain fractional derivative operator, Kragujevac J. Math., 35 (2011), №3, 387-398. 4. M.K. Aouf, A.O. Mostafa, H.M. Zayed, Some characterizations of integral operators associated with certain classes of p-valent functions defined by the Srivastava-Saigo-Owa fractional differintegral operator, Complex Anal. Oper. Theory, 10 (2016), №6, 1267-1275. 5. M.K. Aouf, A.O. Mostafa, H.M. Zayed, Subordination and superordination properties of p-valent functions defined by a generalized fractional differintegral operator, Quaest. Math., 39 (2016) №4, 545-560. 6. M.K. Aouf, A.O. Mostafa, H.M. Zayed, On certain subclasses of multivalent functions defined by a generalized fractional differintegral operator, Afr. Mat., 28 (2017), №1-2, 99-107. 7. T. Bulboaca, Classes of first-order differential superordinations, Demonstr. Math., 35 (2002), №2, 287-292. 8. T. Bulboaca, A class of superordination-preserving integral operators, Indag. Math. (N.S.), 13 (2002), №3, 301-311. 9. T. Bulboaca, Differential subordinations and superordinations, new results, House of Scientific Boook Publ., Cluj-Napoca, 2005. 10. G.P. Goyal, J.K. Prajapat, A new class of analytic p-valent functions with negative coefficients and fractional calculus operators, Tamsui Oxf. J. Inf. Math. Sci., 20 (2004), №2, 175-186. 11. S.S. Miller, P.T. Mocanu, Differential subordinations: theory and applications, Series on Monographs and Textbooks in Pure and Appl. Math., V.255, Marcel Dekker, Inc., New York, 2000. 12. S.S. Miller, P.T. Mocanu, Subordinations of differential superordinations, Complex Var., 48 (2003) №10, 815-826. 13. A.O. Mostafa, M.K. Aouf, H.M. Zayed, T. Bulboac.a, Multivalent functions associated with Srivastava- Saigo-Owa fractional differintegral operator. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, https://doi.org/10.1007/s13398-017-0436-1, /to appear/ 14. S. Owa, On the distortion theorems I. Kyungpook Math. J., 18 (1978), 53-59. 15. W.C. Royster, On the univalence of a certain integral, Michigan Math. J., 12 (1965), №4, 385-387. 16. T.N. Shanmugam, V. Ravichandran, S. Sivasubramanian, Differential sandwich theorems for some subclasses of analytic functions, Aust. J. Math. Anal. Appl., 3 (2006), №1, 1-11.
Pages	52-66
Volume	49
Issue	1
Year	2018
Journal	Matematychni Studii
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Table of content of issue	html