Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 16 (2021), 327 -- 337

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

ON UNIFIED SUBCLASS OF COMPLEX ORDER CONNECTED WITH q-CONFLUENT HYPERGEOMETRIC DISTRIBUTION

Sheza M. El-Deeb

Abstract. In this paper, we apply the concept of q-confluent hypergeometric distribution to introduce and study a unified subclass of univalent functions of complex order ℬζ, γ λ, q(Υ; b, c, m)  consisting of all analytic functions, and some known consequences of the results are also derived for this class.

2020 Mathematics Subject Classification: Primary 05A30, 30C45; Secondary 11B65, 47B38
Keywords: q-Calculus; confluent hypergeometric series;complex order;analytic function.

Full text

References

  1. M. H. Abu Risha, M. H. Annaby, M. E. H. Ismail and Z. S. Mansour, Linear q-difference equations, Z. Anal. Anwend., 26(2007), 481--494. MR2341770.

  2. T. Bulboacă, Differential Subordinations and Superordinations. Recent Results, House of Scientific Book Publ., Cluj-Napoca, 2005.

  3. S. M. El-Deeb, T. Bulboacă and B. M. El-Matary, Maclaurin Coefficient Estimates of Bi-Univalent Functions Connected with the q-Derivative , Mathematics, 8(2020), 1-14, https://doi.org/10.3390/math8030418.

  4. G. Gasper and M. Rahman, Basic hypergeometric series (with a Foreword by Richard Askey). Encyclopedia of mathematics and its applications, Cambridge University Press, Cambridge, 35(1990). MR1052153.

  5. F. H. Jackson, On q-functions and a certain difference operator, Trans. Royal Soc. Edinburgh, 46(2)(1909), 253--281, https://doi.org/10.1017/S0080456800002751.

  6. F. H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math., 41(1910), 193-203. JFM 41.0317.04.

  7. W. C. Ma, D. Minda, A unified treatment of some special classes of univalent functions, in: Proceeding of the conference on complex analysis (Tianjin, 1992), Internat. Press, Cambridge, MA, pp. 157-169. MR1343506. Zbl 1094.30016.

  8. 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 Inc., New York and Basel, 2000. MR1760285. Zbl 0954.34003.

  9. S. Porwal, Confluent hypergeometric distribution and its applications on certain classes of univalent functions of conic regions, Kyungpook Math. J., 58(2018), 495-505. MR3875004. Zbl 1422.30027.

  10. S. Porwal and S. Kumar, Confluent hypergeometric distribution and its applications on certain classes of univalent functions, Afr. Mat., 28(2017), 1-8. MR3613614. Zbl 1368.30008.

  11. E. D. Rainville, Special functions, The Macmillan Co., New York, 1960. MR0107725.

  12. V. Ravichandran, Y. Polatoglu, M. Bolcal and A. Sen, Certain subclasses of starlike and convex functions of complex order, Hacettepe J. Math. Stat. 34 (2005), 9-15. MR2212704.

  13. S. Ruscheweyh, Convolutions in Geometric Function Theory,Presses Univ. Montreal, Montreal, Quebec, 1982. MR674296.

  14. H. M. Srivastava, Certain q-polynomial expansions for functions of several variables. I and II, IMA J. Appl. Math. 30(1983), 205-209. MR767521. Zbl 0544.33007.

  15. H. M. Srivastava, Univalent functions, fractional calculus, and associated generalized hypergeometric functions, in Univalent Functions, Fractional Calculus, and Their Applications (H. M. Srivastava and S. Owa, Editors), Halsted Press (Ellis Horwood Limited, Chichester), pp. 329-354, John Wiley and Sons, New York, Chichester, Brisbane and Toronto, (1989). MR1199160. Zbl 0693.30013.

  16. H. M. Srivastava, Operators of basic (or q-) calculus and fractional q-calculus and their applications in Geometric Function theory of Complex Analysis, Iran J Sci Technol Trans Sci 44(2020), 327--344. MR4064730.

  17. H. M. Srivastava and S. M. El-Deeb, A certain class of analytic functions of complex order with a q-analogue of integral operators, Miskolc Math Notes, 21(2020), no. 1, 417--433. MR4133288. Zbl 07254908.

  18. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian hypergeometric series, Wiley, New York, (1985). MR834385.



Sheza M. El-Deeb
Department of Mathematics,
Faculty of Science, Damietta University,
New Damietta 34517, Egypt.

current address:
Department of Mathematics,
College of Science and Arts, Al- Badaya,
Qassim University, Buraidah 51452, Saudi Arabia.
e-mail: shezaeldeeb@yahoo.com

http://www.utgjiu.ro/math/sma