[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/BKMS.b171122

THE SHARP BOUND OF THE THIRD HANKEL DETERMINANT FOR SOME CLASSES OF ANALYTIC FUNCTIONS

Kowalczyk, Bogumila (Department of Complex Analysis Faculty of Mathematics and Computer Science University of Warmia and Mazury in Olsztyn)
Lecko, Adam (Department of Complex Analysis Faculty of Mathematics and Computer Science University of Warmia and Mazury in Olsztyn)
Lecko, Millenia (Rzeszow University of Technology Faculty of Mathematics and Applied Physics Department of Nonlinear Analysis)
Sim, Young Jae (Department of Mathematics Kyungsung University)

Publication Information

Bulletin of the Korean Mathematical Society / v.55, no.6, 2018 , pp. 1859-1868 More about this Journal

Abstract

In the present paper, we have proved the sharp inequality ${\mid}H_{3,1}(f){\mid}{\leq}4$ and ${\mid}H_{3,1}(f){\mid}{\leq}1$ for analytic functions f with $a_n:=f^{(n)}(0)/n!$ , $n{\in}{\mathbb{N}},$ , such that $$$Re\frac{f(z)}{z}$ > ${\alpha},\;z{\in}{\mathbb{D}}:=\{z{\in}{\mathbb{C}}:{\mid}z{\mid}$ < $1\}$$$ for ${\alpha}=0$ and ${\alpha}=1/2$ , respectively, where $$$H_{3,1}(f):=\left|{\array{{\alpha}_1$ $&$ ${\alpha}_2$ $&$ ${\alpha}_3\\{\alpha}_2$ $&$ ${\alpha}_3$ $&$ ${\alpha}_4\\{\alpha}_3$ $&$ ${\alpha}_4$ $&$ ${\alpha}_5}}\right|$$$ is the third Hankel determinant.

Keywords

univalent functions; $Carath{\acute{e}}odory$ functions; Hankel determinant;

Citations & Related Records

Reference

1	K. O. Babalola, On $H_3(1)$ Hankel determinants for some classes of univalent functions, Inequality Theory and Applications, vol. 6, (Edited by Y. J. Cho), 1-7, Nova Science Publishers, New York, 2010.
2	D. Bansal, S. Maharana, and J. K. Prajapat, Third order Hankel determinant for certain univalent functions, J. Korean Math. Soc. 52 (2015), no. 6, 1139-1148. DOI
3	C. Caratheodory, Uber den Variabilitatsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen, Math. Ann. 64 (1907), no. 1, 95-115. DOI
4	N. E. Cho, B. Kowalczyk, O. S. Kwon, A. Lecko, and Y. J. Sim, Some coefficient inequalities related to the Hankel determinant for strongly starlike functions of order alpha, J. Math. Inequal. 11 (2017), no. 2, 429-439.
5	N. E. Cho, B. Kowalczyk, O. S. Kwon, A. Lecko, and Y. J. Sim, The bounds of some determinants for starlike functions of order alpha, Bull. Malays. Math. Sci. Soc. 41 (2018), no. 1, 523-535. DOI
6	P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer-Verlag, New York, 1983.
7	M. Elin and D. Shoikhet, Linearization Models for Complex Dynamical Systems, Operator Theory: Advances and Applications, 208, Birkhauser Verlag, Basel, 2010.
8	A. W. Goodman, Univalent Functions, Mariner, Tampa, Florida, 1983.
9	A. Janteng, S. A. Halim, and M. Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal. (Ruse) 1 (2007), no. 13-16, 619-625.
10	B. Kowalczyk, A. Lecko, and Y. J. Sim, The sharp bound of the Hankel determinant of the third kind for convex functions, Bull. Austr. Math. Soc., doi: 10.1017/S0004972717001125 (accepted). DOI
11	B. Kowalczyk, A. Lecko, and Y. J. Sim, The sharp bound of the Hankel determinant of the third kind for starlike functions of order 1/2 (submitted).
12	O. S. Kwon, A. Lecko, and Y. J. Sim, On the fourth coefficient of functions in the Caratheodory class, Comput. Methods Funct. Theory, doi: 10.1007/s40315-017-0229-8 (accepted). DOI
13	S. K. Lee, V. Ravichandran, and S. Supramaniam, Bounds for the second Hankel determinant of certain univalent functions, J. Inequal. Appl. 2013 (2013), 281, 17 pp. DOI
14	R. J. Libera and E. J. Zlotkiewicz, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85 (1982), no. 2, 225-230. DOI
15	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), no. 2, 251-257. DOI
16	T. H. MacGregor, The radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 14 (1963), 514-520. DOI
17	A. Marx, Untersuchungen uber schlichte Abbildungen, Math. Ann. 107 (1933), no. 1, 40-67. DOI
18	S. S. Miller and P. T. Mocanu, Differential Subordinations, Monographs and Textbooks in Pure and Applied Mathematics, 225, Marcel Dekker, Inc., New York, 2000.
19	A. K. Mishra and P. Gochhayat, Second Hankel determinant for a class of analytic functions defined by fractional derivative, Int. J. Math. Math. Sci. 2008 (2008), Art. ID 153280, 10 pp.
20	C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Gottingen, 1975.
21	J. K. Prajapat, D. Bansal, A. Singh, and A. K. Mishr, Bounds on third Hankel determinant for close-to-convex functions, Acta Univ. Sapientiae Math. 7 (2015), no. 2, 210-219. DOI
22	J. S. Ratti, The radius of univalence of certain analytic functions, Math. Z. 107 (1968), 241-248. DOI
23	M. Raza and S. N. Malik, Upper bound of the third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli, J. Inequal. Appl. 2013 (2013), 412, 8 pp. DOI
24	M. I. S. Robertson, On the theory of univalent functions, Ann. of Math. (2) 37 (1936), no. 2, 374-408. DOI
25	G. Shanmugam, B. A. Stephen, and K. O. Babalola, Third Hankel determinant for ${\alpha}$ -starlike functions, Gulf J. Math. 2 (2014), no. 2, 107-113.
26	D. Shoikhet, Rigidity and parametric embedding of semi-complete vector fields on the unit disk, Milan J. Math. 84 (2016), no. 1, 159-202. DOI
27	E. Strohhacker, Beitrage zur Theorie der schlichten Funktionen, Math. Z. 37 (1933), no. 1, 356-380. DOI
28	E. Study, Vorlesungen uber ausgewahlte Gegenstande der Geometrie, Zweites Heft; Konforme Abbildung Einfach-Zusammenhangender Bereiche, Druck und Verlag von B.G. Teubner, Leipzig und Berlin, 1913.
29	T. V. Sudharsan, S. P. Vijayalaksmi, and B. A. Sthephen, Third Hankel determinant for a subclass of analytic functions, Malaya J. Math. 2 (2014), no. 4, 438-444.
30	P. Zaprawa, Third Hankel determinants for subclasses of univalent functions, Mediterr. J. Math. 14 (2017), no. 1, Art. 19, 10 pp.