Bound on H3(1) Hankel determinant for pre-starlike functions of order α.



Analytic function, Toeplitz determinants, Pre-starlike function, Convex function, Upper bound, Second and third Hankel functionals, Positive real function, Convolution


The objective of this paper is to obtain best possible upper bound to the third Hankel determinant for the pre-starlike functions of order α (0 ≤ α ≤ 1), using Toeplitz determinants.

Author Biographies

D. Vamshee Krishna, GITAM University.

Department of Mathematics.

D. Shalini, Sri Venkateswara College of Engineering and Technology.

Department of Mathematics.


R. M. Ali, Coefficients of the inverse of strongly starlike functions, Bull. Malays. Math. Sci. Soc., (2nd Series), 26 (1), pp. 63-71, (2003).

K. O. Babalola, On H3(1) Hankel determinant for some classes of univalent functions, Inequality Theory and Applications, 6, pp. 1-7, (2010).

L. de Branges de Bourcia, A proof of Bieberbach conjecture, Acta Mathematica, 154 (1-2), pp. 137-152, (1985).

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

U. Grenander and G. Szegö, Toeplitz forms and their applications. 2nd ed. New York (NY): Chelsea Publishing Co., (1984).

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

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), pp. 251-257, (1983).

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

Ch. Pommerenke, On the coefficients and Hankel determinants of univalent functions, J. Lond. Math. Soc., s1-41 (1), pp. 111-122, (1966).

St. Ruscheweyh, Linear operators between classes of pre-starlike functions, Comm. Math. Helv., 52, pp. 497-509, (1977).

H. Silverman and E. M. Silvia, Pre-starlike functions with negative coefficients, Int. J. Math. Math. Sci., 2 (3), pp. 427-439, (1979).

B. Simon, Orthogonal polynomials on the unit circle, part 1. Classical theory. Vol. 54, American mathematical society colloquium publications. Providence (RI): American Mathematical Society; (2005).

D. Vamshee Krishna and T. RamReddy, Coefficient inequality for parabolic star like functions of order alpha, Afr. Mat., 27 (1-2), pp. 121-132, (2016).

D. Vamshee Krishna and T. RamReddy, An upper bound to the second Hankel determinant for pre-star like functions of order α, Le Matematiche, 70 (2), pp. 109-122, (2015).

D. Vamshee Krishna and T. RamReddy, Coefficient inequality for certain p− valent analytic functions, Rocky Mountain J. Math., 44 (6), pp. 1941-1959, (2014).



How to Cite

D. V. Krishna and D. Shalini, “Bound on H3(1) Hankel determinant for pre-starlike functions of order α.”, Proyecciones (Antofagasta, On line), vol. 37, no. 2, pp. 305-315, Jun. 2018.