International Journal of Nonlinear Analysis and Applications

Subordination and superordination results of multivalent functions associated with the Dziok-Srivastava operator

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt

2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

3 Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania

Abstract

Using the techniques of the differential subordination and superordination, we derive certain subordination and superordination properties of multivalent functions associated with the Dziok-Srivastava operator.

Keywords

[1] B.C. Carlson and D.B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal. 15 (1984) 737–745.
[2] N.E. Cho, O.H. Kwon and H.M. Srivastava, Inclusion and argument properties for certain subclasses of multivalent functions associated with a family of linear operators, J. Math. Anal. Appl. 292 (2004) 470–483.
[3] J.H. Choi, M. Saigo and H.M. Srivastava, Some inclusion properties of a certain family of integral operators, J. Math. Anal. Appl. 276 (2002) 432–445.
[4] J. Dziok and H.M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput. 103 (1999) 1–13.
[5] J. Dziok and H.M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transforms Spec. Funct. 14 (2003) 7–18.
[6] A. Gangadharan, T.N. Shanmugam and HM. Srivastava, Generalized hypergeometric functions associated with k–uniformly convex functions. Comput. Math. Appl. 44 (2002) 1515–1526.
[7] R.M. Goel and N.S. Sohi, A new criterion for p–valent functions, Proc. Amer. Math. Soc. 78 (1980) 353–357.
[8] Y.E. Hohlov, Operators and operations in the class of univalent functions, Izv. Vyssh. Uchebn. Zaved. Mat. 10 (1978) 83–89 (in Russian).
[9] J.L. Liu, Strongly starlike functions associated with the Dziok–Srivastava operator, Tamkang J. Math. 35 (2004) 37–42.
[10] J.L. Liu and K.I. Noor, Some properties of Noor integral operator, J. Nat. Geometry 21 (2002) 81–90.
[11] J.L. Liu and J. Patel, Certain properties of multivalent functions associated with an extended fractional differintegral operator, Appl. Math. Comput. 203 (2008) 703-713.
[12] J.L. Liu and H.M. Srivastava, Certain properties of the Dziok–Srivastava operator, Appl. Math. Comput. 159 (2004) 485–493.
[13] S.S. Miller and P.T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J. 28 (1981) 157–171.
[14] S.S. Miller and P.T. Mocanu, Differential Subordinations: Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker, New York and Basel, 2000.
[15] S.S. Miller and P.T. Mocanu, Subordinants of differential superordinations, Complex Var. 48 (2003) 815–826.
[16] K.I. Noor, Some classes of p−valent analytic functions defined by certain integral operators, Appl. Math. Comput. 157 (2004) 835–840.
[17] K.I. Noor and M.A. Noor, On integral operators, J. Math. Anal. Appl. 238 (1999) 341–352.
[18] M. Nunokawa, On the theory of multivalent functions, Tsukuba J. Math. 11 (1987) 273–286.
[19] S. Owa, On the distortion theorems I, Kyungpook Math. J. 18 (1978) 53–59.
[20] S. Owa and H.M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canad. J. Math. 39 (1987) 1057–1077.
[21] J. Patel and A.K. Mishra, On certain subclasses of multivalent functions associated with an extended fractional differintegral operator, J. Math. Anal. Appl. 332 (2007) 109–122.
[22] St. Ruscheweyh, New criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975) 109–115.
[23] H. Saitoh, A linear operator and its applications of first order differential subordinations, Math. Japonica 44 (1996) 31–38.
[24] T.N. Shanmugam, V. Ravichandran and S. Sivasubramanian, Differential Sandwich theorems for subclasses of analytic functions, Aust. J. Math. Anal. Appl. 3 (2006) 1-11.
[25] H.M. Srivastava, M.K. Aouf, A certain fractional derivative operator and its applications to a new class of analytic and multivalent functions with negative coefficients I and II, J. Math. Anal. Appl. 171 (1992) 1–1; ibid. 192 (1995) 673–688.
[26] H.M. Srivastava and P.W. Karlsson, Multiple Gaussian hypergeometric series, Halsted Press (Ellis Horwood, Chichester), John Wiley and Sons, New York, 1985.
[27] H.M. Srivastava and J. Patel, Some subclasses of multivalent functions involving a certain linear operator, J. Math. Anal. Appl. 310 (2005) 209–228.
[28] E.T. Whittaker and G.N. Watson, A Course on Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; With an Account of the Principal Transcendental Functions, Fourth Edition (Reprinted), Cambridge University Press, Cambridge, 1927.
International Journal of Nonlinear Analysis and Applications
Volume 12, Issue 1
May 2021
Pages 997-1008
  • Receive Date: 17 February 2019
  • Accept Date: 23 November 2019