An upper bound for third Hankel determinant of starlike functions related to shell-like curves connected with Fibonacci numbers

• Autorzy: Sokół J. , İlhan S. , Güney H. Ö.

Identyfikatory

DOI

10.7862/rf.2018.14

• Języki publikacji: EN

Abstrakty

EN

We investigate the third Hankel determinant problem for some starlike functions in the open unit disc, that are related to shell-like curves and connected with Fibonacci numbers. For this, firstly, we prove a conjecture, posed in [17], for sharp upper bound of second Hankel determinant. In the sequel, we obtain another sharp coefficient bound which we apply in solving the problem of the third Hankel determinant for these functions.

Słowa kluczowe

EN

analytic function; convex function; Fibonacci numbers; Hankel determinant; hell-like curve; starlike function

PL

funkcja analityczna; funkcja wypukła; ciąg Fibonacciego; wyznacznik Hankela; krzywa; funkcja gwiaździsta

• Wydawca: Oficyna Wydawnicza Politechniki Rzeszowskiej
• Czasopismo: Journal of Mathematics and Applications
• Rocznik: 2018
• Tom: Vol. 41
• Strony: 195--206
• Opis fizyczny: Bibliogr. 17 poz.

Twórcy

autor

Sokół J.

• Faculty of Mathematics and Natural Sciences, University of Rzeszów, ul. Prof. Pigonia 1, 35-310 Rzeszów, Poland

autor

İlhan S.

• Department of Mathematics, Faculty of Science, Dicle University, TR-21280 Diyarbakir, Turkey

autor

Güney H. Ö.

• Department of Mathematics, Faculty of Science, Dicle University, TR-21280 Diyarbakir, Turkey

Bibliografia

• [1] K.O. Babalola, On H₃(1) Hankel determinant for some classes of univalent functions, Ineq. Theory Appl. 6 (2007) 1-7.
• [2] D. Bansal, S. Maharana, J.K. Prajapat, Third order Hankel determinant for certain univalent functions, J. Korean Math. Soc. 52 (6) (2015) 1139-1148.
• [3] R. Ehrenborg, The Hankel determinant of exponential polynomials, Amer. Math. Monthly 107 (2000) 557-560.
• [4] M. Fekete, G. Szegö, Eine Bemerkung über ungerade schlichte Funktionen, J. London Math. Soc. 8 (1933) 85-89.
• [5] A. Janteng, S. Halim, M. Darus, Coefficient inequality for a function whose derivative has a positive real part, J. Inequal. Pure Appl. Math. 7 (2) (2006) Article 50.
• [6] A. Janteng, S. Halim, M. Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal. 1 (13) (2007) 619-625.
• [7] F.R. Keogh, E.P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20 (1969) 8-12.
• [8] J.W. Layman, The Hankel transform and some of its properties, J. Integer Sequences 4 (2001) 1-11.
• [9] R.J. Libera, E.J. Z lotkiewicz, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc. 87 (2) (1983) 251-257.
• [10] J.W. Noonan, D.K. Thomas, On the second Hankel determinant of areally mean p-valent functions, Trans. Amer. Math. Soc. 223 (2) (1976) 337-346.
• [11] K.I. Noor, Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roum. Math. Pures Appl. 28 (8) (1983) 731-739.
• [12] Ch. Pommerenke, Univalent Functions, Vandenhoeck und Ruprecht, Göttingen, 1975.
• [13] R.K. Raina, J. Sokół, Fekete-Szegö problem for some starlike functions related to shell-like curves, Math. Slovaca 66 (2016) 135-140.
• [14] V. Ravichandran, S. Verma, Bound for the fifth coefficient of certain starlike functions, C. R. Acad. Sci. Paris, Ser. I 353 (6) (2015) 505-510.
• [15] M. Raza, 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) Article 412.
• [16] J. Sokół, On starlike functions connected with Fibonacci numbers, Folia Scient. Univ. Tech. Resoviensis 175 (1999) 111-116.
• [17] J. Sokół, S. İlhan, H. Ö. Güney, Second Hankel determinant problem for several classes of analytic functions related to shell-like curves connected with Fibonacci numbers, TWMS Journal of Applied and Engineering Mathematics 8 (1a) (2018) 220-229.

Uwagi

PL

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).