The sharp bound of the third Hankel determinant for certain subfamilies of analytic functions
Abstract
UDC 517.5
We study an upper bound and the sharp bound of the third-order Hankel determinant for some subfamilies of analytic functions for the parameters $\alpha\in(1,4/3]$ and $\alpha=3/2,$ respectively, also proving the recent conjecture for the sharp bounds made in [Virendra Kumar, Sushil Kumar, V. Ravichandran, Third Hankel determinant for certain classes of analytic functions, Mathematical Analysis I: Approximation Theory, February (2020); DOI: 10.1007/978-981-15-1153-019].
References
M. Arif, Mohsan Raza, Huo Tang, Shehzad Hussain, Hassan Khan, Hankel determinant of order three for familiar subsets of analytic functions related with sine function, Open Math., 17, № 1, 1615–1630 (2019). DOI: https://doi.org/10.1515/math-2019-0132
K. O. Babalola, On $H_3 (1)$ Hankel determinant for some classes of univalent functions, Inequal. Theory and Appl., Nova Sci. Publ., New York (2010), p. 1–7.
S. Banga, S. Sivaprasad Kumar, The sharp bounds of the second and third Hankel determinants for the class $SL^*$, Math. Slovaca, 70, № 4, 849–862 (2020); DOI: 10.1515/ms-2017-0398. DOI: https://doi.org/10.1515/ms-2017-0398
D. Breaz, A. Cătaş, L. Cotîrlă, On the upper bound of the third Hankel determinant for certain class of analytic functions related with exponential function, An. Ştiinƫ. Univ. ``Ovidius'' Constanƫa Ser. Mat., 30, 75–89 (2022); https://doi.org/10.2478/auom-2022-0005. DOI: https://doi.org/10.2478/auom-2022-0005
P. L. Duren, Univalent functions, vol. 259 of Grundlehren der Mathematischen Wissenschaften, Springer, New York (1983).
T. Hayami, S. Owa, Generalized Hankel determinant for certain classes, Int. J. Math. Anal., 4, № 52, 2573–2585 (2010).
B. Kowalczyk, A. Lecko, Y. J. Sim, The sharp bound for the Hankel determinant of the third kind for convex functions, Bull. Aust. Math. Soc., 97, № 3, 435–445 (2018). DOI: https://doi.org/10.1017/S0004972717001125
B. Kowalczyk, A. Lecko, M. Lecko, Y. J. Sim, The sharp bound of the third Hankel deternimant for some classes of analytic functions, Bull. Korean Math. Soc., 55, № 6, 1859–1868 (2018); https://doi.org/10.4134/BKMS.b171122.
O. S. Kwon, A. Lecko , Y. J. Sim, The bound of the Hankel determinant of the third kind for starlike functions, Bull. Malays. Math. Sci. Soc., 42, № 2, 767–780 (2019). DOI: https://doi.org/10.1007/s40840-018-0683-0
O. S. Kwon, Y. J. Sim, The sharp bound of the Hankel determinant of the third kind for starlike functions with real coefficients, Mathematics (2019); DOI:10.3390/math7080721. DOI: https://doi.org/10.20944/preprints201907.0200.v1
R. J. Libera, E. J. Zlotkiewicz, Coefficient bounds for the inverse of a function with derivative in $P$, Proc. Amer. Math. Soc., 87, № 2, 251–257 (1983). DOI: https://doi.org/10.1090/S0002-9939-1983-0681830-8
S. Mahmood, H. M. Srivastava, N. Khan, Q. Z. Ahmad, B. Khan, I. Ali, Upper bound of the third Hankel determinant for a subclass of $q$-starlike functions, Symmetry, 11, № 3, Article ID 347 (2019). DOI: https://doi.org/10.3390/sym11030347
J. Nishiwaki, S. Owa, Coefficient inequalities for certain analytic functions, Int. J. Math. and Math. Sci., 29, № 5, 285–290 (2002). DOI: https://doi.org/10.1155/S0161171202006890
H. Orhan, M. Çağlar, L. Cotîrlǎ, Third Hankel determinant for a subfamily of holomorphic functions related with Lemniscate of Bernoulli, Mathematics, 11, № 5 (2023); https://doi.org/10.3390/math11051147. DOI: https://doi.org/10.3390/math11051147
Ch. Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, Göttingen (1975).
Ch. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J. London Math. Soc., 41, № 1, 111–122 (1966). DOI: https://doi.org/10.1112/jlms/s1-41.1.111
J. K. Prajapat, D. Bansal, S. Maharana, Bounds on third Hankel determinant for certain classes of analytic functions, Stud. Univ. Babeş-Bolyai Math., 62, № 2, 183–195 (2017); DOI:10.24193/subbmath.2017.2.05. DOI: https://doi.org/10.24193/subbmath.2017.2.05
B. Rath, K. S. Kumar, D. V. Krishna, G. K. S. Viswanadh, The sharp bound for the third Hankel determinant of the inverse of functions associated with lemniscate of Bernoulli, Asian-Eur. J. Math., https://doi.org/10.1142/S1793557123501267. DOI: https://doi.org/10.1142/S1793557123501267
K. Sanjay Kumar, B. Rath, D. V. Krishna, The sharp bound of the third Hankel determinant for the inverse of bounded turning functions, Contemp. Math. Internet., 4, № 1, 30–41 (2023); https://doi.org/10.37256/cm.4120232183. DOI: https://doi.org/10.37256/cm.4120232183
L. Shi, M. Shutaywi, N. Alreshidi, M. Arif, M. Ghufran, The sharp bounds of the third-order Hankel determinant for certain analytic functions associated with an eight-shaped domain, Fractal and Fract., 6, № 4 (2022); https://doi.org/10.3390/fractalfract6040223. DOI: https://doi.org/10.3390/fractalfract6040223
B. Rath, K. S. Kumar, D. V. Krishna, 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, 45–50 (2022). DOI: https://doi.org/10.30970/ms.58.1.45-50
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. and Oper. Theory (2022); https://doi.org/10.1007/s11785-022-01241-8. DOI: https://doi.org/10.1007/s11785-022-01241-8
B. A. Uralegaddi, M. D. Ganigi, S. M. Sarangi, Univalent functions with positive coefficients, Tamkang J. Math., 25, № 3, 225–230 (1994); DOI: 10.5556/j.tkjm.25.1994.444. DOI: https://doi.org/10.5556/j.tkjm.25.1994.4448
Virendra Kumar, Sushil Kumar, V. Ravichandran, Third Hankel determinant for certain classes of analytic functions, Mathematical Analysis I: Approximation Theory (2020); DOI: 10.1007/978-981-15-1153-019. DOI: https://doi.org/10.1007/978-981-15-1153-0_19
Copyright (c) 2024 Biswajit Rath
This work is licensed under a Creative Commons Attribution 4.0 International License.
