The sharp bound of the third Hankel determinant of the \(k^{th}\)-root transformation for bounded turning functions. (English) Zbl 1537.30015
Summary: The objective of this paper is to estimate the sharp bound of the third Hankel determinant for the \(k^{th}\)-root transformation to the class of functions whose derivative has a positive real part satisfying the normalized conditions \(f(0)=0\) and \(f'(0)=1\) in the open unit disk \(\mathbb{D} := \{z \in \mathbb{C} : |z| < 1\}\).
MSC:
| 30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |
| 30C50 | Coefficient problems for univalent and multivalent functions of one complex variable |
Keywords:
univalent function; Hankel determinant; bounded turning functions; \(k^{th}\)-root transformation; Carathéodory functionReferences:
| [1] | R. M. Ali, S. K. Lee, V. Ravichandran, and S. Supramaniam, The Fekete-Szegö coefficient functional for transforms of analytic functions, Bull. Iran. Math. Soc. 35 (2009), 119-142. · Zbl 1193.30006 |
| [2] | K. O. Babalola, On H3(1) Hankel determinant for some classes of univalent func-tions. In: Inequality Theory and Applications 6 (Y. J. Cho et al., eds), Nova Science Publishers, New York, 2010, 7 pp. |
| [3] | S. Banga and S. Sivaprasad Kumar, The sharp bounds of the second and third Hankel determinants for the class SL*, Math. Slovaca 70 (2020), 849-862. DOI · Zbl 1482.30031 · doi:10.1515/ms-2017-0398 |
| [4] | C. Carathéodory, Über den variabilitätsbereich der fourier’schen konstanten von pos-itiven harmonischen funktionen, Rend. Circ. Matem. Palermo 32 (1911), 193-217. DOI · JFM 42.0429.01 · doi:10.1007/BF03014795 |
| [5] | P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften 259, Springer, New York, 1983. · Zbl 0514.30001 |
| [6] | A. W. Goodman, Univalent Functions, Mariner, Tampa, Florida, 1983. · Zbl 1041.30501 |
| [7] | T. Hayami and S. Owa, Generalized Hankel determinant for certain classes, Int. J. Math. Anal. 4 (2010), 2573-2585. · Zbl 1226.30015 |
| [8] | B. Kowalczyk and A. Lecko, The sharp bound of the third Hankel determinant for functions of bounded turning. Bol. Soc. Mat. Mex. 27 (2021), 69. DOI · Zbl 1476.30055 · doi:10.1007/s40590-021-00383-7 |
| [9] | B. Kowalczyk, A. Lecko and Young Jae Sim, The sharp bound for the Hankel deter-minant of the third kind for convex functions, Bull. Austral. Math. Soc. 97 (2018), 435-445. · Zbl 1394.30007 |
| [10] | O. S. Kwon, A. Lecko, and Y. J. Sim, The bound of the Hankel determinant of the third kind for starlike functions, Bull. Malays. Math. Sci. Soc. 42 (2019), 767-780. · Zbl 1419.30007 |
| [11] | O. S. Kwon, A. Lecko, and Y. J. Sim, On the fourth coefficient of functions in the Carathéodory class, Comput. Methods Funct. Theory 18 (2018), 307-314. · Zbl 1395.30019 |
| [12] | R. J. Libera and E. J. Zlotkiewicz, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85 (1982), 225-230. · Zbl 0464.30019 |
| [13] | R. J. Libera and E. J. Zlotkiewicz, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc. 87 (1983), 251-257. · Zbl 0488.30010 |
| [14] | Ch. Pommerenke, Univalent Functions, Vandenhoeck and Ruprecht, Göttingen, 1975. |
| [15] | Ch. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J. Lond. Math. Soc. 41 (1966), 111-122. · Zbl 0138.29801 |
| [16] | B. Rath, K. S. Kumar, N. Vani, and 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 Math. J. (Accepted) · Zbl 1543.30045 |
| [17] | 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 (2022), 45-50. · Zbl 1507.30005 |
| [18] | B. Rath, K. S. Kumar, D. V. Krishna, and A. Lecko, The sharp bound of the third Hankel determinant for starlike functions of order 1/2, Complex Anal. Oper. Theory, 8 pp. (2022). DOI · Zbl 1493.30036 · doi:10.1007/s11785-022-01241-8 |
| [19] | K. Sanjay Kumar, B Rath, and D. V. Krishna. The sharp bound of the third Hankel determinant for the inverse of bounded turning functions, Contemp. Math. 4 (2023), 30-41. DOI · doi:10.37256/cm.4120232183 |
| [20] | L. Shi, H. M. Srivastava, A. Rafiq, M. Arif, and M. Ihsan, Results on Hankel deter-minants for the inverse of certain analytic functions subordinated to the exponential function, Mathematics 10 (2022), 3429, 15 pp. DOI · doi:10.3390/math10193429 |
| [21] | Y. J. Sim and Pawel Zaprawa, Third Hankel determinants for two classes of analytic functions with real coefficients, Forum Math. 33 (2021), 973-986. DOI · Zbl 1486.30051 · doi:10.1515/forum-2021-0014 |
| [22] | 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. · Zbl 1498.30014 |
| [23] | H. M. Srivastava, G. Murugusundaramoorthy, and T. Bulboacȃ,The second Hankel de-terminant for subclasses of bi-univalent functions associated with a nephroid domain, Rev. Real Acad. Cienc. Exactas Fís. Natur. Ser. A Mat. (RACSAM) 116 (2022), 145, 21 pp. DOI · Zbl 1496.30012 · doi:10.1007/s13398-022-01286-6 |
| [24] | H. M. Srivastava, S. Kumar, V. Kumar, and N. E. Cho, Hermitian-Toeplitz and Han-kel determinants for starlike functions associated with a rational function, J. Nonlinear Convex Anal. 23 (2022), 2815-2833. · Zbl 1499.30172 |
| [25] | H. M. Srivastava, T. G. Shaba, G. Murugusundaramoorthy, A. K. Wanas, and G. I. Oros, The Fekete-Szegö functional and the Hankel determinant for a certain class of analytic functions involving the Hohlov operator, AIMS Math. 8 (2023), 340-360. |
| [26] | G. K. Surya Viswanadh, B. Rath, K. Sanjay Kumar, and D. Vamshee Krishna, The sharp bound for the third Hankel determinant of the inverse of functions associated with lemniscate of Bernoulli, Asian-Eur. J. Math. 16 (2023), 2350126. DOI · doi:10.1142/S1793557123501267 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
