Korean J. Math. Vol. 33 No. 3 (2025) pp.205-218
DOI: https://doi.org/10.11568/kjm.2025.33.3.205

Bi-Bazilevic functions based on Hurwitz-Lerch Zeta function associated with exponential Pareto distribution

Main Article Content

Murugusundaramoorthy Gangadharan

Abstract

In this paper, we introduce and investigate new subclass of bi-univalent functions defined in the open unit disk, which are based on Hurwitz-Lerch Zeta function associated with exponential Pareto distribution , satisfying subordinate conditions. Furthermore, we find estimates on the Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ for functions in these new subclass. Several new consequences of the results are also pointed out.Additionally we discussed Fekete-Szegö inequality results



Article Details

References

[1] T. Mandal, Certain results on three-dimensional f-Kenmotsu manifolds with conformal Ricci solitons, Korean J. Math. 30 (1) (2022), 1–10. https://doi.org/10.11568/kjm.2022.30.1.1 Google Scholar

[2] Y. G. Wondifraw, Pseudo-complementation on generalized almost distributive fuzzy lattices, Korean J. Math. 30 (1) (2022), 11–23. https://doi.org/10.11568/kjm.2022.30.1.11 Google Scholar

[3] F. Q. Gouvêa, p-adic numbers. An introduction, Springer, 2020. https://link.springer.com/book/10.1007/978-3-030-47295-5 Google Scholar

[4] P. L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften Series, 259, Springer Verlag, New York, 1983. https://link.springer.com/book/9780387907956 Google Scholar

[5] W. C. Ma, D. Minda, A unified treatment of some special classes of functions, in: Proceedings of the Conference on Complex Analysis, Tianjin, 1992, 157–169, Conf. Proc. Lecture Notes Anal. 1. Int. Press, Cambridge, MA, 1994. Google Scholar

[6] J. W. Alexander, Functions which map the interior of the unit circle upon simple regions, Ann. of Math. 17 (1915), 12–22. http://dx.doi.org/10.2307/2007212 Google Scholar

[7] K. A. Al-Kadim, M. A. Boshi, Exponential Pareto Distribution, Math. Theory Model. 3 (2013), 135–146. Google Scholar

[8] R. C. Gupta, R. D. Gupta, P. L. Gupta, Modelling failure time data by Lehman alternatives, Commun. Stat. Theory Methods 27 (1998), 887–904. https://doi.org/10.1080/03610929808832134 Google Scholar

[9] H. Haj Ahmad, E. M. Almetwally, M. Elgarhy, D. A. Ramadan, On unit exponential Pareto distribution for modeling the recovery rate of COVID-19, Processes 11 (2023), 232. https://doi.org/10.3390/pr11010232 Google Scholar

[10] A. O. Idowu, I. A. Ajibode, An exponential-Pareto distribution approach to improving raw material quality, FEPI-JOPAS 5 (2023), 33–39. Google Scholar

[11] A. S. Hassan, S. E. Hemeda, S. G. Nassr, On the extension of exponentiated Pareto Distribution, J. Mod. Appl. Stat. Methods 19 (2020), 21. Google Scholar

[12] I. Elbatal, M. Zayed, M. Rasekhi, N. S. Butt, The Exponential Pareto Power Series Distribution: Theory and Application, Pak. J. Stat. Oper. Res. 13 (2017), 603–615. https://doi.org/10.18187/pjsor.v13i3.2072 Google Scholar

[13] H. M. Srivastava, Min-Jie Luo, R. K. Raina, New results involving a class of generalized Hurwitz-Lerch Zeta functions and their applications, Turkish J. Anal. Number Theory 1 (1) (2013), 26–35. https://doi.org/10.12691/tjant-1-1-7 Google Scholar

[14] I. B. Jung, Y. C. Kim, H. M. 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 Google Scholar

[15] J. Choi, H. M. 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 Google Scholar

[16] 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 Google Scholar

[17] 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. Google Scholar

[18] L. de Branges, A proof of the Bieberbach conjecture, Acta Math. 154 (1985), 137–152. https://doi.org/10.1007/BF02392821 Google Scholar

[19] R. J. Libera, E. J. Zlotkiewicz, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc. 87 (1983), 251–257. Google Scholar

[20] D. A. Brannan, J. Clunie, W. E. Kirwan, Coefficient estimates for a class of star-like functions, Canad. J. Math. 22 (1970), 476–485. https://doi.org/10.4153/CJM-1970-055-8 Google Scholar

[21] D. A. Brannan, T. S. Taha, On some classes of bi-univalent functions, Studia Univ. Babeș-Bolyai Math. 31 (2) (1986), 70–77. https://doi.org/10.1016/B978-0-08-031636-9.50012-7 Google Scholar

[22] M. Lewin, On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc. 18 (1967), 63–68. https://doi.org/10.1090/S0002-9939-1967-0206255-1 Google Scholar

[23] E. Netanyahu, The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in z < 1, Arch. Rational Mech. Anal. 32 (1969), 100–112. https://doi.org/10.1007/BF00247676 Google Scholar

[24] T. S. Taha, Topics in Univalent Function Theory, Ph.D. Thesis, University of London, 1981. Google Scholar

[25] H. M. Srivastava, A. K. Mishra, P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23 (2010), 1188–1192. https://doi.org/10.1016/j.aml.2010.05.009 Google Scholar

[26] E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Classical Analysis 2 (1) (2013), 49–60. Google Scholar

[27] T. Hayami, S. Owa, Coefficient bounds for bi-univalent functions, Pan Amer. Math. J. 22 (4) (2012), 15–26. Google Scholar

[28] G. Murugusundaramoorthy, Coefficient estimates of bi-Bazilevic functions defined by Srivastava-Attiya operator, Le Matematiche 69 (2014), Fasc. II, 43–56. Google Scholar

[29] G. Murugusundaramoorthy, Coefficient estimates of bi-Bazilevic functions of complex order based on quasi subordination involving Srivastava-Attiya operator, Indian J. Math. 58 (3) (2016), 267–286. Google Scholar

[30] H. M. Srivastava, G. Murugusundaramoorthy, K. Vijaya, Coefficient estimates for some families of bi-Bazilevic functions of the Ma-Minda type involving the Hohlov operator, J. Classical Anal. 22 (2013), 167–181. Google Scholar

[31] J. O. Hamzat, M. O. Oluwayemi, A. T. Oladipo, A. A. Lupas, On α-Pseudo spirallike functions associated with exponential Pareto distribution (EPD) and Libera integral operator, Mathematics 12 (9) (2024), 1305. https://doi.org/10.3390/math12091305 Google Scholar

[32] P. Zaprawa, Estimates of initial coefficients for Biunivalent functions, Abstract Appl. Anal., 2014, Art. ID: 357480. https://doi.org/10.1155/2014/357480 Google Scholar

[33] R. Singh, On Bazilevič functions, Proc. Amer. Math. Soc. 28 (1973), 261–271. https://doi.org/10.2307/2040613 Google Scholar

[34] Y. C. Kim, T. Sugawa, A note on Bazilevič functions, Taiwanese J. Math. 13 (2009), 1489–1495. https://doi.org/10.11650/twjm/1500405555 Google Scholar

[35] Z.-G. Wang, H.-T. Wang, A class of multivalent non-Bazilevič functions involving the Cho-Kwon-Srivastava operator, Tamsui Oxford J. Math. Sci. 26 (2010), 1–19. Google Scholar

[36] C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975. Google Scholar

[37] W. Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc. (Ser. 2) 48 (1943), 48–82. https://doi.org/10.1112/plms/s2-48.1.48 Google Scholar

[38] M. Fekete, G. Szegö, Eine Bemerkung über ungerade schlichte funktionen, J. London Math. Soc. s1-8 (1933), 85–89. https://doi.org/10.1112/jlms/s1-8.2.85 Google Scholar

[39] P. Zaprawa, On the Fekete-Szegö problem for classes of bi-univalent functions, Bull. Belg. Math. Soc. Simon Stevin 21 (1) (2014), 1–192. https://doi.org/10.36045/bbms/1394544302 Google Scholar

[40] G. Murugusundaramoorthy, T. Bulboacă, Initial Coefficients and Fekete-Szegö inequalities for functions related to Van Der Pol numbers (VPN), Mathematica Slovaca 73 (5) (2023), 1183–1196. https://doi.org/10.1515/ms-2023-0087 Google Scholar