SECOND HANKEL DETERMINANT PROBLEM FOR SEVERAL CLASSES OF ANALYTIC FUNCTIONS RELATED TO SHELL-LIKE CURVES CONNECTED WITH FIBONACCI NUMBERS

Year 2018, Volume: 8 Issue: 1.1, 220 - 229, 01.09.2018

J. Sokol S. İlhan H.  O. Güney

Abstract

In this paper, we investigate upper bounds for the second Hankel determinant in several classes of analytic functions in the open unit disc, related to shell-like curves and connected with Fibonacci numbers.

Keywords

Analytic functions, shell-like curve, Fibonacci numbers, starlike functions, convex functions, Hankel determinant

References

• Dziok, J., Raina, R. K., Sok´o l, J., 61(2011) 2605-2613, Certain results for a class of convex functions related to a shell-like curve connected with Fibonacci numbers, Comp. Math. Appl.
• Dziok, J., Raina, R. K., Sok´o l, 218 (2011) 996–1002, On α−convex functions related to a shell-like curve connected with Fibonacci numbers, Appl. Math. Comp.
• Ehrenborg, R., 107 (2000): 557–560, The Hankel determinant of exponential polynomials, Amer. Math. Monthly.
• Fekete, M., Szeg¨o, G., Eine Bemerkung, 8 (1933) 85–89, ¨Uber ungerade schlichte Functionen, J. London Math. Soc.
• Janteng, A., Halim, S., Darus, M., 7 (2) (2006), Article 50, Coefficient inequality for a function whose derivative has a positive real part, J. Inequal. Pure Appl. Math.
• Janteng, A., Halim, S., Darus, M., 1 (13) (2007) 619–625, Hankel determinant for starlike and convex functions, Int. J. Math. Anal.
• Keogh, F. R., Merkes,E. P., 20 (1969) 8-12, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc.
• Layman, J. W., 4 (2001): 1-11. The Hankel transform and some of its properties, J. Integer Sequences
• Libera, R. J., Z lotkiewicz, E. J., 87 (2) (1983) 251-257, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc.
• Noonan, J. W., Thomas, D. K., 223 (2) (1976) 337-346, On the second Hankel determinant of a really mean p-valent functions, Trans. Amer. Math. Soc.
• Noor, K. I., 28 (8) (1983) 731-739, Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roum. Math. Pures Appl.
• Pommerenke, Ch., 1975, Univalent Functions, Math. Math, Lehrbucher, Vandenhoeck and Ruprecht, G¨ottingen.
• Raina, R. K., Sok´o l, J., 66 (2016) 135-140, Fekete-Szeg¨o problem for some starlike functions related to shell-like curves, Math. Slovaca.
• Sok´o l, J., 175 (1999) 111-116, On starlike functions connected with Fibonacci numbers, Folia Scient.