The sharp bound of certain second Hankel determinants for inverse functions of the class of convex functions with respect to symmetric points

Vijaya Kumar Ch.; Biswajit Rath; SANJAY KUMAR K.; VAMSHEE KRISHNA D.

doi:10.5269/bspm.64878

Ch. Vijaya Kumar
Biswajit Rath GITAM University https://orcid.org/0000-0002-9146-6628
K. SANJAY KUMAR
D. VAMSHEE KRISHNA

Abstract

In this paper, we focus on the sharp upper bounds for specific second-order Hankel determinants pertaining to the inverse function, denoted as f−1. Our investigation is centered on scenarios where f is a member of the class of convex functions with respect to symmetric points. By delving into this particular class, we aim to uncover and establish sharp bounds for these second Hankel determinants, providing valuable insights to extremal functions within this context.

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References

R. M. Ali,Coefficient of the inverse of strongly starlike functions,Bull. Malayasian Math. Sci. Soc. (second series) 26(2003), 63-71.

R. N. Das, P. Singh, On subclass of schlicht mappings, Indian J. Pure and Appl. Math., 8 (1977), 864-872.

P. L. Duren, Univalent functions, Vol. 259 of Grundlehren der Mathematischen Wissenschaften, Springer, New York, USA, 1983.

H. O. Guney, G. Murugusundaramoorthy and H. M. Srivastava, The second Hankel determinant for a certain class of bi-close-to-convex functions, Results Math. 74 (2019), Article ID 93, 1–13.

T. Hayami and S. Owa, Generalized Hankel determinant for certain classes, Int. J. Math. Anal., 4(52)(2010), 2573–2585.

A. L. P. Hern, A. Janteng and R. Omar, Hankel Determinant H2(3) for Certain Subclasses of Univalent Functions, Mathematics and Statistics 8(5), 566-569, 2020, doi: 10.13189/ms.2020.080510.

A. Janteng, S. A. Halim and M. Darus, Hankel Determinant for starlike and convex functions, Int. J. Math. Anal., 1(13)(2007), 619–625.

A. Janteng, S. A. Halim and M. Darus, Coefficient inequality for a function whose derivative has a positive real part, J. Inequal. Pure Appl. Math., 7(2)(2006), 1–5.

R. J. Libera, E. J. Zlotkiewicz, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85 (1982), no. 2, 225–230.

R. J. Libera and E. J. Zlotkiewicz, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc., 87(2)(1983), 251–257.

Ch. Pommerenke, Univalent functions, Gottingen: Vandenhoeck and Ruprecht; 1975.

Ch. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J. Lond. Math. Soc., 41(s-1) (1966), 111–122.

T. Ramreddy and D. Vamshee Krishna, Hankel determinant for stalike and convex with respect to symmetric points, Journal of the Indian Math. Soc., (2012), 161-171.

B. Rath, K. Sanjay Kumar, D. Vamshee Krishna, CH. Vijay Kumar, N. Vani, Hankel determinant of certain order for bounded turning functions of order alpha. Indian Journal Math., 64(3) (2022), 401–416.

B. Rath, K. S. Kumar, D. V. Krishna, A. Lecko, The sharp bound of the third Hankel determinant for starlike functions of order 1/2, Complex Anal. Oper. Theory (2022), https://doi.org/10.1007/s11785-022-01241-8.

B. Rath, K. S. Kumar, D. V. Krishna and G. K. S. Viswanadh, The sharp bound of the third Hankel determinants for inverse of starlike functions with respect to symmetric points, Mat. Stud. 58 (1) (2022), 45–50.

B. Rath, K. S. Kumar and D. V. Krishna, An exact estimate of the third Hankel determinants for functions inverse to convex functions, Mat. Stud. 60 (1) (2023), 34–39.

B. Rath, K. S. Kumar, N. Vani, D. V. Krishna, The Sharp Bound of Certain Second Hankel Determinants for the Class of Inverse of Starlike Functions With Respect to Symmetric Points. Ukrainian Mathematical Journal, 75 (10) (2023), DOI: 10.3842/umzh.v75i10.7255.

B. Rath, K. Sanjay Kumar, D. Vamshee Krishna, CH. Vijay Kumar, N. Vani, Fifth hankel determinant for multivalent bounded turning function of order α. Journal of the Indian Math. Soc., 90(3–4) (2023), 289–308.

L. Shi, H. M. Srivastava, M. Arif, S. Hussain and H. Khan, An investigation of the third Hankel determinant problem for certain subfamilies of univalent functions involving the exponential function, Symmetry 11 (2019), Article ID 598, 1–14.

L. Shi, H. M. Srivastava, A. Rafiq, M. Arif and M. Ihsan, Results on Hankel determinants for the inverse of certain analytic functions subordinated to the exponential function, Mathematics 10 (2022), Article ID 3429, 1–15.

Y. J. Sim, A. Lecko and D. K. Thomas, The second Hankel determinant for strongly convex and Ozaki close-to-convex functions ,Annali di Matematica Pura ed Applicata, (2021), 2515–2533,https://doi.org/10.1007/s10231-021-01089-3.

H. M. Srivastava, A. K. Mishra, and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Applied Mathematics Letters, vol. 23, no. 10, pp. 1188–1192, 2010.

H. M. Srivastava, Q. Z. Ahmad, M. Darus, N. Khan, B. Khan, N. Zaman and H. H. Shah, Upper bound of the third Hankel determinant for a subclass of close-to-convex functions associated with the lemniscate of Bernoulli, Mathematics 7 (2019), Article ID 848, 1–10.

H. M. Srivastava, G. Kaur and G. Singh, Estimates of the fourth Hankel determinant for a class of analytic functions with bounded turnings involving cardioid domains, J. Nonlinear Convex Anal. 22 (2021), 511–526.

H. M. Srivastava, B. Khan, Nazar Khan, M. Tahir, S. Ahmad and Nasir Khan, Upper bound of the third Hankel determinant for a subclass of q-starlike functions associated with the q-exponential function, Bull. Sci. Math. 167 (2021), Article ID 102942, 1–18.

H. M. Srivastava, S. Kumar, V. Kumar and N. E. Cho, Hermitian-Toeplitz and Hankel determinants for starlike functions associated with a rational function, J. Nonlinear Convex Anal. 23 (2022), 2815–2833.

H. M. Srivastava, G. Murugusundaramoorthy and T. Bulboac˘a, The second Hankel determinant for subclasses of bi-univalent functions associated with a nephroid domain, Rev. Real Acad. Cienc. Exactas F´ıs. Natur. Ser. A Mat. (RACSAM) 116 (2022), Article ID 145, 1–21.

H. M. Srivastava, T. G. Shaba, G. Murugusundaramoorthy, A. K. Wanas and G. I. Oros, The Fekete-Szego functional and the Hankel determinant for a certain class of analytic functions involving the Hohlov operator, AIMS Math. 8 (2022), 340–360.

P. Zaprawa, On Hankel Determinant H2(3) for Univalent Functions, Results Math (2018), https://doi.org/10.1007/s00025-018-0854-1. .