Abstract
In this paper, we investigate the upper bound associated with the second Hankel determinant \(H_{2}(2)\) for a certain class of bi-close-to-convex functions which we have introduced here. Several closely related results are also considered.
Similar content being viewed by others
References
Altınkaya, Ş., Yalçın, S.: Coefficient estimates for two new subclasses of bi-univalent functions with respect to symmetric points. J. Funct. Spaces, 1–5 (2015)
Babalola, K.O.: On \(H_{3}(1)\) Hankel determinant for some classes of univalent functions. In: Cho, Y.J., Kim, J.K., Dragomir, S.S. (eds.) Inequality Theory and Applications, vol. 6, pp. 1–7. Nova Science Publishers, Hauppauge (2010)
Branges, D.L.: A proof of the Bieberbach conjecture. Acta Math. 154, 137–152 (1985)
Brannan D.A., Clunie J.G.: Aspects of contemporary complex analysis. In: Proceedings of the NATO Advanced Study Institute Held at the University of Durham. Academic Press, New York (1980)
Carathéodory, C.: Über den variabilitägatsbereich der Fourierschen Konstanten Von Positiven harmonischen Funktionen. Rend. Circ. Mat. Palermo (Ser. 1) 32, 193–217 (1911)
Deniz, E., Çaglar, M., Orhan, H.: Second Hankel determinant for bi-starlike and bi-convex functions of order \(\beta \). Appl. Math. Comput. 271, 301–307 (2015)
Duren P.L.: Univalent Functions, In: Grundlehren der Mathematischen Wissenschaften, Band 259. Springer-Verlag, New York, Berlin, Heidelberg, Tokyo (1983)
Janteng, A., Halim, S., Darus, M.: Coefficient inequality for a function whose derivative has a positive real part. J. Inequal. Pure Appl. Math. 7, 1–5 (2006)
Janteng, A., Halim, S., Darus, M.: Hankel Determinant for starlike and convex functions. Int. J. Math. Anal. 1, 619–625 (2007)
Kedzierawski, A.W.: Some remarks on bi-univalent functions. Ann. Univ. Mariae Curie-Sklodowska Sect. A 39, 77–81 (1985)
Keogh, F.R., Merkes, E.P.: A coefficient inequality for certain classes of analytic functions. Proc. Am. Math. Soc. 20, 8–12 (1969)
Lewin, M.: On a coefficient problem for bi-univalent functions. Proc. Am. Math. Soc. 18, 63–68 (1967)
Libera, R.J., Zlotkiewicz, E.J.: Coefficient bounds for the inverse of a function with derivative in \({\cal{P}}\). Proc. Am. Math. Soc. 87, 251–257 (1983)
Netanyahu, E.: The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z|\(< 1\). Arch. Ration. Mech. Anal. 32, 100–112 (1969)
Orhan, H., Magesh, N., Yamini, J.: Bounds for the second Hankel determinant of certain bi-univalent functions. Turk. J. Math. 40, 679–687 (2016)
Srivastava, H.M., Bulut, S., Çağlar, M., Yağmur, N.: Coefficient estimates for a general subclass of analytic and bi-univalent functions. Filomat 27, 831–842 (2013)
Srivastava, H.M., Gaboury, S., Ghanim, F.: Initial coefficient estimates for some subclasses of \(m\)-fold symmetric bi-univalent functions. Acta Math. Sci. Ser. B Engl. Ed. 36, 863–871 (2016)
Srivastava, H.M., Mishra, A.K., Das, M.K.: The Fekete–Szegö problem for a subclass of close-to-convex functions. Complex Var. Theory Appl. 44, 145–163 (2001)
Srivastava, H.M., Mishra, A.K., Gochhayat, P.: Certain subclasses of analytic and bi-univalent functions. Appl. Math. Lett. 23, 1188–1192 (2010)
Srivastava, H.M., Sümer, Eker S., Ali, R.M.: Coefficient Bounds for a certain class of analytic and bi-univalent functions. Filomat 29, 1839–1845 (2015)
Tan, D.L.: Coefficient estimates for bi-univalent functions. Chin. Ann. Math. Ser. A 5, 559–568 (1984)
Tang, H., Srivastava, H.M., Sivasubramanian, S., Gurusamy, P.: The Fekete–Szegö functional problems for some classes of \(m\)-fold symmetric bi-univalent functions. J. Math. Inequal. 10, 1063–1092 (2016)
Xu, Q.-H., Gui, Y.-C., Srivastava, H.M.: Coefficient estimates for a certain subclass of analytic and bi-univalent functions. Appl. Math. Lett. 25, 990–994 (2012)
Xu, Q.-H., Xiao, H.-G., Srivastava, H.M.: A certain general subclass of analytic and bi-univalent functions and associated coefficient estimate problems. Appl. Math. Comput. 218, 11461–11465 (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Güney, H.Ö., Murugusundaramoorthy, G. & Srivastava, H.M. The Second Hankel Determinant for a Certain Class of Bi-Close-to-Convex Functions. Results Math 74, 93 (2019). https://doi.org/10.1007/s00025-019-1020-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-019-1020-0