Application of neutrosophic Poisson distribution series on harmonic classes of analytic functions defined by q− derivative operator and sigmoid function

Abstract

There are several authors who have obtained various forms of properties for some subclasses of analytic univalent functions related to different distribution series, such as Binomial, Generalized Discrete Probability, Geometric, Mittag-Leffler, Pascal, and Poisson distribution series. The authors, in this paper, proved the inclusion relation of the harmonic analytic function class $H_{q}^{\alpha}(\theta, \gamma(s), \Psi)$ established by applying convolution operators regarding neutrosophic distribution series equipped with the Sigmoid function (activation function). The present results are capable of handling both accurate (determinate) data and inaccurate (indeterminate) data.

Keywords

• Analytic
• harmonic starlike
• harmonic convex
• neutrosophic Poisson distribution
• q-derivative

References

1. Ahuja, O. P., Jahangiri, J. M., A subclass of harmonic univalent functions, J. Nat. Geom., 20 (2001), 45–56.
2. Ahuja, O. P., C¸ etinkaya, A., Use of quantum calculus approach in mathematical sciences and its role in geometric function theory, AIP Conference Proceedings, 2095(1) (2019), 1-14. https://doi.org/10.1063/1.5097511
3. Ahuja, O. P., Jahangiri, J. M., Noshiro-type harmonic univalent functions, Sci. Math. Jpn., 6 (2002), 253–259.
4. Alhabib, R., Ranna, M. M., Farah, H., Salama, A. A., Some nuetrosophic probability distributions, Neutrosophic Sets Syst., 22 (2018), 30–37.
5. Awolere, I. T., Oladipo, A. T., Application of neutrosophic Poisson probability distribution series for certain subclass of analytic univalent function, TWMS J. App. and Eng. Math., 13(3) (2023), 1042-1052.
6. Awolere, I. T., Hankel determinant for bi-Bazelevic function involing error and sigmoid function defined by derivative calculus via Chebyshev polynomials, J. Frac. Calc. Appl., 11(2) (2020), 208-217.
7. Aydogan, M., Bshouty, D., Lyzzaik, A., Sakar, F. M., On the shears of univalent harmonic mappings, Complex Anal. Oper. Theory, 13 (2019), 2853-2862. https://doi.org/10.1007/s11785-018-0855-9
8. Bayram, H., q-Analogue of a new subclass of harmonic univalent functions associated with subordination, Symmetry, 14 (2022), 1-15. https://doi.org/10.3390/sym14040708

9. Bshouty, D., Lyzzaik, A., Sakar, F. M., Harmonic mappings of bounded boundary ratation, Proc. Am. Math. Soc., 146 (2018), 1113-1221. http://dx.doi.org/10.1090/proc/13796
10. Chandrashekar, R., Lee, S. K., Subramanian, K. G., Hyergeometric functions and subclasses of harmonic mappings, Proceeding of the International Conference on Mathematical Analysis, 2010, Bangkok, 2010, 95–103.
11. Clunie, J., Sheil-Small, T., Harmonic univalent functions, Ann. Acad. Sci. Fen., 9 (1984), 3–25.
12. Duren, P., Harmonic Mappings in Plane, Cambridge Tracts in Mathematics; Cambridge University Press: Cambridge UK, 156, 2004.
13. Frasin, B. A., Comprehensive family of harmonic univalent functions, SUT J. Math., 42 (2006), 145-155. http://dx.doi.org/10.55937/sut/1159988041
14. Frasin, B. A., Alb Lupas, A., An application of Poisson distribution series on harmonic classes of analytic functions, Symmetry, 15(3) (2023), 1-11. https://doi.org/10.3390/sym15030590
15. Jackson, F. H., On q-definite integrals, Quart. J. Pure Appl. Math., 14 (1910), 193-203. https://doi.org/10.1080/14786447108640600
16. Jackson, F.H., On q-functions and a certain difference operator, Earth Environ. Sci. Trans. R. Soc. Edinb., 46(2) (1908), 253–281. https://doi.org/10.1017/S0080456800002751
17. Jahangiri, J. M., Harmonic functions starlike in the unit disk, J. Math. Anal. Appl., 235(2) (1999), 470-477. https://doi.org/10.1006/jmaa.1999.6377
18. Murugusundaramoorthy, G., Vijaya, K., Frasin, B. A., A subclass of harmonic funtion with negative coefficients defined by Dziok-Srivastava operator, Tamkang J. Math., 42(4) (2011), 463-473. https://doi.org/10.5556/j.tkjm.42.2011.231
19. Oladipo, A. T., Gbolagade, A. M., Some subordination results for logistic sigmoid activation function in the space of univalent functions, Adv. Comput. Sci. Eng., 12 (2014), 61-79.
20. Oladipo, A. T., Coefficient inequality for a subclass of analytic univalent functions related to simple logistic functions, Stud. Univ. Babes-Bolyai Math., 61(1) (2016), 45-52.
21. Oladipo, A. T., Bounds for Poisson and neutrosophic Poisson distribution associated with Chebyshev polynomials, Palest. J. Math., 10(1) (2019), 169–174.
22. Porwal, S., Srivastava, D., Some connections between various classes of planar harmonic mappings involving Poisson distribution series, Electronic J. Math. Anai. Appl., 6(2) (2018), 163-171.
23. Silverman, H., Harmonic univalent function with negative coefficients, J. Math. Anal. Appl., 220(1) (1998), 283–289. https://doi.org/10.1006/jmaa.1997.5882
24. Silverman, H., Silvia, E. M., Subclasses of harmonic univalent functions, New Zealand J. Math., 28 (1999), 275-284.
25. Smarandache, F., Khalid, H. E., Neutrosophic Precalculus and Neutrosophic Calculus: Neutrosophic Applications, Infinite Study, PONS, Stuttgart, Germany, 2nd edition, 2018.
26. Sokol, J., Ibrahim, R. W., Ahmad, M. Z., Al-Janaby, H. F., Inequalities of harmonic univalent functions with connections of hypergeometric functions, Open Math., 13 (2015), 691–705. https://doi.org/10.1515/math-2015-0066
27. Srivastava, H. M., Khan, N. Khan, S., Ahmad, Q. Z., Khan, B., A Class of k-symmetric harmonic functions involving a certain q-derivative operator, Mathematics, 9(15) (2021), 1-14. https://doi.org/10.3390/math9151812
28. Yalçın, S., Öztürk, M., Yamankaradeniz, M., A subclass of harmonic univalent functions with negative coefficients, Appl. Math. Comput., 142(2-3) (2003), 469-–476. https://doi.org/10.1016/S0096-3003(02)00314-4
29. Yalçın, S., Öztürk, M., A new subclass of complex harmonic functions, Math. Inequal. Appl., 7(1) (2004), 55–61.
30. Yousef, A. T., Salleh, Z., On a harmonic univalent subclass of functions involving a generalized linear operator, Axioms, 9(1) (2020), 1-10. https://doi.org/10.3390/axioms9010032