Certain Subclass of Analytic Functions Defined by Bell Distribution Series

Main Article Content

G. Ramu, J. Srinivas, P. Thirupathi Reddy, K. K. Viswanathan, H. Niranjan

Abstract

The study of the geometric properties of analytic functions and their numerous applications in a variety of mathematical fields, including fractional calculus, probability distributions, and special functions, has drawn significant and impressive attention to Geometric Function Theory (GFT), one of the most prominent branches of complex analysis, in recent years. The focus of this article is to introduce a new subclass of analytic functions involving Bell Distribution series and obtain coefficient inequalities, neighborhood results and partial sums for this class.

Article Details

References

  1. J.W. Alexander, Functions Which Map the Interior of the Unit Circle Upon Simple Regions, Ann. Math. 17 (1915), 12–22. https://doi.org/10.2307/2007212.
  2. A. Amourah, T. Al-Hawary, F. Yousef, J. Salah, Collection of Bi-Univalent Functions Using Bell Distribution Associated with Jacobi Polynomials, Int. J. Neutrosophic Sci. 25 (2025), 228–238.
  3. A. Amourah, M. Alomari, F. Yousef, A. Alsoboh, Consolidation of a Certain Discrete Probability Distribution with a Subclass of Bi-Univalent Functions Involving Gegenbauer Polynomials, Math. Probl. Eng. 2022 (2022), 6354994. https://doi.org/10.1155/2022/6354994.
  4. A. Amourah, B.A. Frasin, M. Ahmad, F. Yousef, Exploiting the Pascal Distribution Series and Gegenbauer Polynomials to Construct and Study a New Subclass of Analytic Bi-Univalent Functions, Symmetry 14 (2022), 147. https://doi.org/10.3390/sym14010147.
  5. E. Aqlan, J.M. Jahangiri, S.R. Kulkarni, New Classes of $k$-Uniformly Convex and Starlike Functions, Tamkang J. Math. 35 (2004), 261–266. https://doi.org/10.5556/j.tkjm.35.2004.207.
  6. L. Bain, M. Engelhardt, Introduction to Probability and Mathematical Statistics, Duxbury Press, 1992.
  7. E.T. Bell, Exponential Polynomials, Ann. Math. 35 (1934), 258–277. https://doi.org/10.2307/1968431.
  8. E.T. Bell, Exponential Numbers, Am. Math. Mon. 41 (1934), 411–419. https://doi.org/10.1080/00029890.1934.11987615.
  9. S.D. Bernardi, Convex and Starlike Univalent Functions, Trans. Am. Math. Soc. 135 (1969), 429–446. https://doi.org/10.2307/1995025.
  10. F. Castellares, S.L. Ferrari, A.J. Lemonte, On the Bell Distribution and Its Associated Regression Model for Count Data, Appl. Math. Model. 56 (2018), 172–185. https://doi.org/10.1016/j.apm.2017.12.014.
  11. J. Choi, H. Srivastava, Certain Families of Series Associated with the Hurwitz–Lerch Zeta Function, Appl. Math. Comput. 170 (2005), 399–409. https://doi.org/10.1016/j.amc.2004.12.004.
  12. C. Ferreira, J.L. López, Asymptotic Expansions of the Hurwitz–Lerch Zeta Function, J. Math. Anal. Appl. 298 (2004), 210–224. https://doi.org/10.1016/j.jmaa.2004.05.040.
  13. T. Flett, The Dual of an Inequality of Hardy and Littlewood and Some Related Inequalities, J. Math. Anal. Appl. 38 (1972), 746–765. https://doi.org/10.1016/0022-247x(72)90081-9.
  14. M. Garg, K. Jain, H.M. Srivastava, Some Relationships Between the Generalized Apostol–Bernoulli Polynomials and Hurwitz–Lerch Zeta Functions, Integral Transform. Spec. Funct. 17 (2006), 803–815. https://doi.org/10.1080/10652460600926907.
  15. A.W. Goodman, Univalent Functions and Nonanalytic Curves, Proc. Am. Math. Soc. 8 (1957), 598–601. https://doi.org/10.2307/2033525.
  16. A. Goodman, On Uniformly Starlike Functions, J. Math. Anal. Appl. 155 (1991), 364–370. https://doi.org/10.1016/0022-247x(91)90006-l.
  17. I. Jung, Y. Kim, H. Srivastava, The Hardy Space of Analytic Functions Associated with Certain One-Parameter Families of Integral Operators, J. Math. Anal. Appl. 176 (1993), 138–147. https://doi.org/10.1006/jmaa.1993.1204.
  18. S. Lin, H. Srivastava, Some Families of the Hurwitz–Lerch Zeta Functions and Associated Fractional Derivative and Other Integral Representations, Appl. Math. Comput. 154 (2004), 725–733. https://doi.org/10.1016/s0096-3003(03)00746-x.
  19. S. Lin, H.M. Srivastava, P. Wang, Some Expansion Formulas for a Class of Generalized Hurwitz–Lerch Zeta Functions, Integral Transform. Spec. Funct. 17 (2006), 817–827. https://doi.org/10.1080/10652460600926923.
  20. S. Owa, T. Sekine, R. Yamakawa, On Sakaguchi Type Functions, Appl. Math. Comput. 187 (2007), 356–361. https://doi.org/10.1016/j.amc.2006.08.133.
  21. J.K. Prajapat, S.P. Goyal, Applications of Srivastava-Attiya Operator to the Classes of Strongly Starlike and Strongly Convex Functions, J. Math. Inequal. 3 (2009), 129–137. https://doi.org/10.7153/jmi-03-13.
  22. D. Răducanu, H.M. Srivastava, A New Class of Analytic Functions Defined by Means of a Convolution Operator Involving the Hurwitz–Lerch Zeta Function, Integral Transform. Spec. Funct. 18 (2007), 933–943. https://doi.org/10.1080/10652460701542074.
  23. S. Ruscheweyh, Neighborhoods of Univalent Functions, Proc. Am. Math. Soc. 81 (1981), 521–521. https://doi.org/10.1090/s0002-9939-1981-0601721-6.
  24. K. Sakaguchi, On a Certain Univalent Mapping., J. Math. Soc. Jpn. 11 (1959), 72–75. https://doi.org/10.2969/jmsj/01110072.
  25. S.M. Popade, R.N. Ingle, P.T. Reddy, B. Venkateswarlu, A New Subclass of Analytic Functions Defined by Linear Operator, Adv. Math.: Sci. J. 9 (2020), 205–217. https://doi.org/10.37418/amsj.9.1.17.
  26. H. Silverman, Partial Sums of Starlike and Convex Functions, J. Math. Anal. Appl. 209 (1997), 221–227. https://doi.org/10.1006/jmaa.1997.5361.
  27. E.M. Silvia, Partial Sums of Convex Functions of Order $alpha$, Houston J. Math. 11 (1985), 397–404.
  28. H.M. Srivastava, A.A. Attiya, An Integral Operator Associated with the Hurwitz–Lerch Zeta Function and Differential Subordination, Integral Transform. Spec. Funct. 18 (2007), 207–216. https://doi.org/10.1080/10652460701208577.
  29. H.M. Srivastava, J. Choi, Series Associated with the Zeta and Related Functions, Springer, 2001. https://doi.org/10.1007/978-94-015-9672-5.