Skip to main content
SpringerLink
Log in

An application of generalized distribution series on certain subclasses of univalent functions

  • Original Article
  • Published:
Boletín de la Sociedad Matemática Mexicana Aims and scope Submit manuscript

Abstract

In this study, we establish a relation between particular subclasses of complex-order univalent functions and generalized distribution series. The main aim of the present investigation is to obtain the necessary and sufficient conditions for generalized distribution series \({\mathcal {P}}_{\psi }(z)\) belong to the classes \({\mathcal {S}}(\delta ,\eta ,\beta )\) and \({\mathcal {R}}(\delta ,\eta ,\beta )\). We also provide some properties of integral operator connected to generalized distribution series and sufficient conditions for \({\mathcal {F}}_{\psi }(z)\) to be in \({\mathcal {K}}(\eta ,\delta )\) and \({\mathcal {R}}(\eta ,\delta )\). Also we discuss some corollaries and consequences of the major results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abdul Halim, S.: On a class of functions of complex order. Tamkang J. Math. 30(2), 147–153 (1999)

  2. Altınkaya, Ş, Yalçın, S.: Poisson distribution series for analytic univalent functions. Complex Anal. Oper. Theory 12(5), 1315–1319 (2018)

  3. Altıntaş, O., Özkan, Ö.: Starlike, convex and close-to-convex functions of complex order. Hacet. Bull. Nat. Sci. Eng. Ser. B 28, 37–46 (1999)

    MathSciNet  Google Scholar 

  4. Altıntaş, O., Özkan, Ö., Srivastava, H.M.: Neighborhoods of a class of analytic functions with negative coefficients. Appl. Math. Lett. 13(3), 63–67 (2000)

    Article  MathSciNet  Google Scholar 

  5. Altunhan, A., Eker, S.S.: On certain subclasses of univalent functions of complex order associated with Poisson distribution series. Bol. Soc. Mat. Mex. (3) 26(3), 1023–1034 (2020)

    Article  MathSciNet  Google Scholar 

  6. Aouf, M.K.: Some subordinations results for certain subclasses of starlike and convex functions of complex order. Acta Univ. Apulensis Math. Inform. No. 35, 101–110 (2013)

    MathSciNet  Google Scholar 

  7. Bansal, D., Prajapat, J.K.: Certain geometric properties of the Mittag-Leffler functions. Complex Var. Elliptic Equ. 61(3), 338–350 (2016)

    Article  MathSciNet  Google Scholar 

  8. Baricz, A.: Generalized Bessel Functions of the First Kind. Lecture Notes in Mathematics, vol. 1994. Springer-Verlag, Berlin (2010)

    Book  Google Scholar 

  9. Bohra, N., Ravichandran, V.: On confluent hypergeometric functions and generalized Bessel functions. Anal. Math. 43(4), 533–545 (2017)

    Article  MathSciNet  Google Scholar 

  10. Dixit, K.K., Pal, S.K.: On a class of univalent functions related to complex order. Indian J. Pure Appl. Math. 26(9), 889–896 (1995)

    MathSciNet  Google Scholar 

  11. El-Ashwah, R.M., Kota, W.Y.: Some condition on a Poisson distribution series to be in subclasses of univalent functions. Acta Univ. Apulensis Math. Inform. No. 51, 89–103 (2017)

    MathSciNet  Google Scholar 

  12. El-Deeb, S.M., Bulboacă, T., Dziok, J.: Pascal distribution series connected with certain subclasses of univalent functions. Kyungpook Math. J. 59(2), 301–314 (2019)

    MathSciNet  Google Scholar 

  13. El-Deeb, S.M., Bulboacă, T.: Differential sandwich-type results for symmetric functions associated with Pascal distribution series. J. Contemp. Math. Anal. 56(4), 214–224 (2021)

    Article  MathSciNet  Google Scholar 

  14. Frasin, B.A.: On certain subclasses of analytic functions associated with Poisson distribution series. Acta Univ. Sapientiae Math. 11(1), 78–86 (2019)

    MathSciNet  Google Scholar 

  15. Gangadharan, A., Shanmugam, T.N., Srivastava, H.M.: Generalized hypergeometric functions associated with \(k\)-uniformly convex functions. Comput. Math. Appl. 44(12), 1515–1526 (2002)

    Article  MathSciNet  Google Scholar 

  16. Lashin, A.M.Y., Badghaish, A.O., Bajamal, A.Z.: Certain subclasses of univalent functions involving Pascal distribution series. Bol. Soc. Mat. Mex. 28, 12 (2022)

    Article  MathSciNet  Google Scholar 

  17. Murugusundaramoorthy, G., A Frasin, B., Al-Hawary, T.: Uniformly convex spiral functions and uniformly spirallike function associated with Pascal distribution series. arXiv arXiv:2001.07517 (2020)

  18. Murugusundaramoorthy, G., Vijaya, K., Porwal, S.: Some inclusion results of certain subclass of analytic functions associated with Poisson distribution series. Hacet. J. Math. Stat. 45(4), 1101–1107 (2016)

  19. Nasr, M.A., Aouf, M.K.: Starlike function of complex order. J. Nat. Sci. Math. 25(1), 1–12 (1985)

    MathSciNet  Google Scholar 

  20. Nazeer, W., Mehmood, Q., Kang, S.M., Ul-Haq, A.: An application of binomial distribution series on certain analytic functions. J. Comput. Anal. Appl. 26(1), 11–17 (2019)

    MathSciNet  Google Scholar 

  21. Owa, S.: Notes on starlike, convex, and close-to-convex functions of complex order, in Univalent functions, fractional calculus, and their applications (Koriyama) 199–218. Ellis Horwood Ser. Math, Appl, Horwood, Chichester (1988)

  22. Ponnusamy, S., Rønning, F.: Starlikeness properties for convolutions involving hypergeometric series. Ann. Univ. Mariae Curie-Skłodowska Sect. A 52(1), 141–155 (1998)

    MathSciNet  Google Scholar 

  23. Porwal, S.: An application of a Poisson distribution series on certain analytic functions. J. Complex Anal. 2014, 984135 (2014)

  24. Porwal, S.: Generalized distribution and its geometric properties associated with univalent functions. J. Complex Anal. 2018, 8654506 (2018)

  25. Porwal, S., Gupta, A.: Some properties of convolution for hypergeometric distribution type series on certain analytic univalent functions. Acta Univ. Apulensis Math. Inform. No. 56, 69–80 (2018)

    MathSciNet  Google Scholar 

  26. Porwal, S., Kumar, S.: Confluent hypergeometric distribution and its applications on certain classes of univalent functions. Afr. Mat. 28(1–2), 1–8 (2017)

    Article  MathSciNet  Google Scholar 

  27. Raina, R.K.: On univalent and starlike Wright’s hypergeometric functions. Rend. Sem. Mat. Univ. Padova 95, 11–22 (1996)

    MathSciNet  Google Scholar 

  28. Silverman, H.: Starlike and convexity properties for hypergeometric functions. J. Math. Anal. Appl. 172, 574–581 (1993)

    Article  MathSciNet  Google Scholar 

  29. Srivastava, H.M., Murugusundaramoorthy, G., Sivasubramanian, S.: Hypergeometric functions in the parabolic starlike and uniformly convex domains. Integral Transforms Spec. Funct. 18(7–8), 511–520 (2007)

    Article  MathSciNet  Google Scholar 

  30. Sümer Eker, S., Murugusundaramoorthy, G., Şeker, B., Çekiç, B.: Spiral-like functions associated with Miller-Ross-type Poisson distribution series. Bol. Soc. Mat. Mex. 29, 16 (2023)

    Article  MathSciNet  Google Scholar 

  31. Swaminathan, A.: Certain sufficiency conditions on Gaussian hypergeometric functions. JIPAM. J. Inequal. Pure Appl. Math. 5(4), 10 (2004). (Article 83)

  32. Vanitha, L., Ramachandran, C., Bulboaca, T.: Certain subclasses of spirallike univalent functions related to Poisson distributions series. Turk. J. Math. 45, 1449–1458 (2021)

    Article  MathSciNet  Google Scholar 

  33. Wiatrowski, P.: The coefficients of a certain family of holomorphic functions, Zeszyty Nauk. Uniw. Łódz. Nauki Mat. Przyrod. Ser. II No. 39 Mat., 75–85 (1971)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Arts and Sciences, University of Bursa Uludag, Görükle, Bursa, Turkey

    Sibel Yalçın

  2. Department of Mathematics, Vel Tech High Tech Dr. Rangarajan Dr. Sakunthala Engineering College, Avadi, Chennai, 600062, Tamilnadu, India

    Krishnan Marimuthu

  3. Department of Mathematics, SRM Institute of Science and Technology, Kattankulathur, Chennai, 603203, Tamilnadu, India

    Jayaraman Uma

Authors
  1. Sibel Yalçın
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Krishnan Marimuthu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jayaraman Uma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sibel Yalçın.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yalçın, S., Marimuthu, K. & Uma, J. An application of generalized distribution series on certain subclasses of univalent functions. Bol. Soc. Mat. Mex. 30, 3 (2024). https://doi.org/10.1007/s40590-023-00575-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40590-023-00575-3

Keywords

Mathematics Subject Classification

Search

Navigation