The sharp bound of the third Hankel determinant for functions of bounded turning. (English) Zbl 1476.30055
Summary: We find the sharp bound for the third Hankel determinant
\[
H_{3,1} (f):= \begin{vmatrix} a_1 & a_2 & a_3 \\ a_2 & a_3 & a_4 \\ a_3 & a_4 & a_5
\end{vmatrix}
\]
for analytic functions \(f\) with \(a_n :=f^{(n)} (0)/n!\), \(n\in \mathbb{N}\), \(a_1 :=1\), such that
\[
\mathrm{Re}\, f^{\prime} (z)>0,\quad z\in \mathbb{D} :=\{ z\in \mathbb{C} : |z|<1\}.
\]
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 |
References:
| [1] | Alexander, JW, Functions which map the interior of the unit circle upon simple regions, Ann. Math., 17, 1, 12-22 (1915) · JFM 45.0672.02 · doi:10.2307/2007212 |
| [2] | Babalola, KO, On \(H_3(1)\) Hankel determinant for some classes of univalent functions, Inequal. Theory Appl., 6, 1-7 (2010) |
| [3] | Carathéodory, C., Über den Variabilitatsbereich der Koeffizienten von Potenzreihen, die gegebene werte nicht annehmen, Math. Ann., 64, 95-115 (1907) · JFM 38.0448.01 · doi:10.1007/BF01449883 |
| [4] | Carlitz, L., Hankel determinants and Bernoulli numbers, Tohoku Math. J., 5, 272-276 (1954) · Zbl 0055.27003 · doi:10.2748/tmj/1178245272 |
| [5] | Cho, NE; Kowalczyk, B.; Kwon, OS; Lecko, A.; Sim, YJ, The bounds of some determinants for starlike functions of order alpha, Bull. Malays. Math. Sci. Soc., 41, 523-535 (2018) · Zbl 1387.30007 · doi:10.1007/s40840-017-0476-x |
| [6] | Goodman, AW, Univalent Functions (1983), Tampa: Mariner, Tampa · Zbl 1041.30501 |
| [7] | Hayman, WK, On the second Hankel determinant of mean univalent functions, Proc. Lond. Math. Soc., 18, 3, 77-94 (1968) · Zbl 0158.32101 · doi:10.1112/plms/s3-18.1.77 |
| [8] | Janteng, A.; Halim, SA; Darus, M., Coefficient inequality for a function whose derivative has a positive real part, J. Inequal. Pure Appl. Math., 7, 2, 1-5 (2006) · Zbl 1134.30310 |
| [9] | Janteng, A.; Maharana, S.; Prajapat, JK, Third order Hankel determinant for certain univalent functions, J. Korean Math. Soc., 52, 6, 1-5 (2015) · Zbl 1328.30005 |
| [10] | Kanas, A.; Lecko, A., On the Fekete-Szegö problem and the domain of convexity for a certain class of univalent functions, Folia Sci. Univ. Tech. Resov., 73, 49-57 (1990) · Zbl 0741.30012 |
| [11] | Kowalczyk, B.; Lecko, A.; Lecko, M.; Sim, YJ, The sharp bound of the third Hankel determinant for some classes of analytic functions, Bull Korean Math. Soc., 55, 6, 1859-1868 (2018) · Zbl 1411.30017 |
| [12] | Kowalczyk, B.; Lecko, A.; Sim, YJ, The sharp bound of the Hankel determinant of the third kind for convex functions, Bull. Aust. Math. Soc., 97, 435-445 (2018) · Zbl 1394.30007 · doi:10.1017/S0004972717001125 |
| [13] | Kwon, OS; Lecko, A.; Sim, YJ, On the fourth coefficient of functions in the Carathéodory class, Comput. Methods Funct. Theory, 18, 307-314 (2018) · Zbl 1395.30019 · doi:10.1007/s40315-017-0229-8 |
| [14] | Lecko, A.; Sim, YJ; Śmiarowska, B., The sharp bound of the Hankel determinant of the third kind for starlike functions of order 1/2, Complex Anal. Oper. Theory, 13, 2231-2238 (2019) · Zbl 1422.30022 · doi:10.1007/s11785-018-0819-0 |
| [15] | Lee, SK; Ravichandran, V.; Supramanian, S., Bound for the second Hankel determinant of certain univalent functions, J. Inequal. Appl., 2013, 281, 1-17 (2013) · Zbl 1302.30018 |
| [16] | Libera, RJ; Zlotkiewicz, EJ, Early coefficients of the inverse of a regular convex function, Proc. Am. Math. Soc., 85, 2, 225-230 (1982) · Zbl 0464.30019 · doi:10.1090/S0002-9939-1982-0652447-5 |
| [17] | Libera, RJ; Zlotkiewicz, EJ, Coefficient bounds for the inverse of a function with derivatives in \({\cal{P}} \), Proc. Am. Math. Soc., 87, 2, 251-257 (1983) · Zbl 0488.30010 |
| [18] | MacGregor, TH, Functions whose derivative has a positive real part, Trans. Am. Math. Soc., 104, 4, 532-537 (1962) · Zbl 0106.04805 · doi:10.1090/S0002-9947-1962-0140674-7 |
| [19] | Obradović, M.; Tuneski, N., Some properties of the class \(\cal{U} \), Ann. Univ. Mariae Curie-Skłodowska Sect. A, 73, 49-56 (2019) · Zbl 1433.30042 |
| [20] | Pommerenke, C., On the coefficients and Hankel determinants of univalent functions, J. Lond. Math. Soc., 41, 111-122 (1966) · Zbl 0138.29801 · doi:10.1112/jlms/s1-41.1.111 |
| [21] | Pommerenke, C., On the Hankel determinants of Univalent functions, Mathematika, 14, 108-112 (1967) · Zbl 0165.09602 · doi:10.1112/S002557930000807X |
| [22] | Pommerenke, C., Univalent Functions (1975), Göttingen: Vandenhoeck & Ruprecht, Göttingen · Zbl 0298.30014 |
| [23] | Schoenberg, IJ, On the maxima of certain Hankel determinants and the zeros of the classical orthogonal polynomials, Indag. Math., 21, 282-290 (1959) · Zbl 0090.04401 · doi:10.1016/S1385-7258(59)50031-1 |
| [24] | Zaprawa, P., Determinants, third Hankel, for subclasses of univalent functions, Mediterr. J. Math., 14, 1, 1-10 (2017) · Zbl 1362.30027 · doi:10.1007/s00009-016-0829-y |
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.
