Article ID: MTJPAM-D-20-00044

Title: Subclasses of Starlike and Convex Functions Associated with Mittag-Leffler-type Poisson Distribution Series


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-20-00044; Volume 3 / Issue 3 / Year 2021 (Special Issue), Pages 147-154

Document Type: Research Paper

Author(s):   Basem Aref Frasin a , Saurabh Porwal b ,   Feras Yousef c

aFaculty of Science, Department of Mathematics, Al al-Bayt University, Mafraq, Jordan

bDepartment of Mathematics, Ram Sahai Government Degree College, Bairi-Shivrajpur, Kanpur-209205,(U.P.), India

cUniversity of Jordan, Department of Mathematics, 11942 Amman, Jordan

Received: 29 November 2020, Accepted: 31 January 2021, Published: 25 April 2021.

Corresponding Author: Basem Aref Frasin (Email address: drbasem64@gmail.com)

Full Text: PDF

Abstract

In this paper, we find the necessary and sufficient conditions, inclusion relations for Mittag-Leffler-type Poisson distribution series belonging to the classes  \mathcal{S}^{\mathcal{\ast }}(\zeta ,\delta )  and  \mathcal{C}^{\mathcal{\ast }}(\zeta ,\delta ). Further, we consider an integral operator related to Mittag-Leffler-type Poisson distribution series.

Keywords: Analytic functions, Univalent functions, Mittag-Leffler-type Poisson distribution series

References:
  1. M. K. Aouf, A generalization of multivalent functions with negative coefficients, J. Korean Math. Soc. 25 (1), 53-66, 1988.
  2. M. K. Aouf and T. M. Seoudy, Subclasses of p-valent functions involving a new operator containing the generalized Mittag–Leffler function, Mediterr. J. Math. 15, Article number: 181, 2018, https://doi.org/10.1007/s00009-018-1228-3.
  3. M. K. Aouf and T. M. Seoudy, Some preserving sandwich results of certain operator containing a generalized Mittag-Leffler function, Bol. Soc. Mat. Mex. 25, 577-588, 2019, https://doi.org/10.1007/s40590-018-0224-8.
  4. A. A. Attiya, Some Applications of Mittag-Leffler Function in the Unit Disk, Filomat 30 (7), 2075-2081, 2016.
  5. D. Bansal and J. K. Prajapat, Certain geometric properties of the Mittag-Leffler functions, Complex Var. Elliptic Equ. 61 (3), 338-350, 2016.
  6. A. Baricz, Geometric properties of generalized Bessel functions, Publ. Math. Debrecen 73 (1-2), 155-178, 2008.
  7. N. E. Cho, S. Y. Woo and S. Owa, Uniform convexity properties for hypergeometric functions, Fract. Cal. Appl. Anal. 5 (3), 303-313, 2002.
  8. H. E. Darwish, A. Y. Lashin and E. M. Madar, On subclasses of uniformly starlike and convex functions de ned by Struve functions, Int. J. Open Problems Complex Analysis 8 (1), 34-43, 2016.
  9. K. K. Dixit and S. K. Pal, On a class of univalent functions related to complex order, Indian J. Pure Appl. Math. 26 (9), 889-896, 1995.
  10. R. M. El-Ashwah and W. Y Kota, Some condition on a Poisson distribution series to be in subclasses of univalent functions, Acta Univ. Apulensis Math. Inform. 51/2017, pp. 89-103.
  11. S. M. El-Deeb, T. Bulboacă and J. Dziok, Pascal distribution series connected with certain subclasses of univalent functions, Kyungpook Math. J. 59, 301-314, 2019.
  12. B. A. Frasin, An application of an operator associated with generalized Mittag-Leffler function, Konuralp J. Math. 7 (1), 199-202, 2019.
  13. B. A. Frasin, On certain subclasses of analytic functions associated with Poisson distribution series, Acta Univ. Sapientiae, Mathematica 11 (1), 78-86, 2019.
  14. B. A. Frasin, Subclasses of analytic functions associated with Pascal distribution series, Adv. Theory Nonlinear Anal. Appl. 4 (2), 92-99, 2020.
  15. B. A. Frasin and I. Aldawish, On subclasses of uniformly spirallike functions associated with generalized Bessel functions, J. Funct. Spaces, 2019, Article ID 1329462, 6 pages, 2019.
  16. B. A. Frasin and M. M. Gharaibeh, Subclass of analytic functions associated with Poisson distribution series, Afr. Mat. 2020, https://doi.org/10.1007/s13370-020-00788-z.
  17. B. A. Frasin, T. Al-Hawary and F. Yousef, Necessary and sufficient conditions for hypergeometric functions to be in a subclass of analytic functions, Afr. Mat. 30, 223-230, 2019, https://doi.org/10.1007/s13370-018-0638-5.
  18. B. A. Frasin, T. Al-Hawary and F. Yousef, Some properties of a linear operator involving generalized Mittag-Leffler function, Stud. Univ. Babeş-Bolyai Math. 65 (1), 67-75, 2020.
  19. M. Garg, P. Manohar and S. L. Kalla, A Mittag-Leffler-type function of two variables, Integral Transforms Spec. Funct. 24 (11), 934-944, 2013.
  20. V. P. Gupta and P. K. Jain, Certain classes of univalent functions with negative coefficents, Bull. Aust. Math. Soc. 14, 409-416, 1976.
  21. T. Janani and G. Murugusundaramoorthy, Inclusion results on subclasses of starlike and convex functions associated with Struve functions, Italian J. Pure Applied Math. 32, 467-476, 2014.
  22. V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics Series, 301. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1994.
  23. N. Magesh, S. Porwal and C. Abirami, Starlike and convex properties for Poisson distribution series, Stud. Univ. Babeş-Bolyai Math. 63 (1), 71-78, 2018.
  24. E. Merkes and B. T. Scott, Starlike hypergeometric functions, Proc. Amer. Math. Soc. 12, 885-888, 1961.
  25. G. M. Mittag-Leffler, Sur la nouvelle fonction E(x), C. R. Acad. Sci. Paris, 137, 554-558, 1903.
  26. S. R. Mondal and A. Swaminathan, Geometric properties of generalized Bessel functions, Bull. Malays. Math. Sci. Soc. 35 (1), 179-194, 2012.
  27. A. O. Mostafa, A study on starlike and convex properties for hypergeometric functions, J. Inequal. Pure Appl. Math. 10 (3), 1-16, 2009.
  28. G. Murugusundaramoorthy, Subclasses of starlike and convex functions involving Poisson distribution series, Afr. Mat. 28, 1357-1366, 2017, https://doi.org/10.1007/s13370-017-0520-x.
  29. G. Murugusundaramoorthy, B. A. Frasin and T. Al-Hawary, Uniformly convex spiral functions and uniformly spirallike function associated with Pascal distribution series, arXiv:2001.07517 [math.CV].
  30. G. Murugusundaramoorthy, K. Vijaya and S. Porwal, Some inclusion results of certain subclass of analytic functions associated with Poisson distribution series, Hacet. J. Math. Stat. 45 (4), 1101-1107, 2016.
  31. S. Porwal, An application of a Poisson distribution series on certain analytic functions, J. Complex Anal. 2014, Article ID: 984135, 1-3, 2014.
  32. S. Porwal and K. K. Dixit, On Mittag-Leffler type Poisson distribution, Afr. Mat. 28, 29-34, 2017, https://doi.org/10.1007/s13370-016-0427-y.
  33. S. Porwal, N.  Magesh and C. Abirami, Certain subclasses of analytic functions associated with Mittag–Leffler-type Poisson distribution series, Bol. Soc. Mat. Mex. 2020, https://doi.org/10.1007/s40590-020-00288-x.16.
  34. H. Silverman, Univalent functions with negative coefficents, Proc. Amer. Math. Soc. 51, 109-116, 1975.
  35. H. Silverman, Starlike and convexity properties for hypergeometric functions, J. Math. Anal. Appl. 172, 574-581, 1993.
  36. H. M. Srivastava, B. A. Frasin and V. Pescar, Univalence of integral operators involving Mittag-Leffler functions, Appl. Math. Inf. Sci. 11 (3), 635-641, 2017.
  37. A. Wiman, Über die Nullstellun der Funcktionen E(x), Acta Math. 29, 217-134, 1905.