Abstract
In this paper, we introduce a subclass of analytic and bi-univalent functions in the open unit disk. Here, we give upper bound estimates for the second Hankel determinant of the functions that belong to this class. Some interesting applications and conclusions of the results obtained in this paper are also discussed.
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Brannan, D.A., Taha, T.S.: On some classes of bi-univalent functions. Stud. Univ. Babes-Bolyai Math. 31, 70–77 (1986)
Deniz, E., Çağlar, M., Orhan, H.: Second Hankel determinant for bi-stalike and bi-convex functions of order \(\beta \). Appl. Math. Comput. 271, 301–307 (2015)
Duren, P.L.: Univalent Functions (Grundlehren der Mathematischen Wissenschaften, 259, New York). Springer, New York (1983)
Fekete, M., Szegö, G.: Eine Bemerkung über ungerade schichte Funktionen. J. Lond. Math. Soc. 8(8), 85–89 (1933)
Goodman, A.W.: Univalent Functions. Volume I. Polygonal, Washington (1983)
Grenander, U., Szegö, G.: Toeplitz Forms and Their Applications. California Monographs in Mathematical Sciences. University of California Press, Berkeley, CA, USA (1958)
Hummel, J.: The coefficient regions of starlike functions. Pacific J. Math. 7, 1381–1389 (1957)
Hummel, J.: Extremal problems in the class of starlike functions. Proc. Am. Math. Soc. 11, 741–749 (1960)
Kanas, S.: An unified approach to the FeketeSzeg problem. Appl. Math. Comput. 218(17), 8453–8461 (2012)
Kanas, S., Lecko, A., Stankiewicz, J.: Differential subordinations and geometric means. Complex Var Theory Appl Int J 28(3), 201–209 (1996)
Kanas, S., Kim, S.-A., Sivasubramanian, S.: Verification of Brannan and Clunies conjecture for certain subclasses of bi-univalent function. Ann. Polon. Math. 113, 295–304 (2015)
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(1), 63–68 (1967)
Libera, R.J., Złotkiewicz, E.J.: Early coefficients of the inverse of a regular convex function. Proc. Am. Math. Soc. 85(2), 225–230 (1982)
Libera, R.J., Złotkiewicz, E.J.: Coefficient bounds for the inverse of a function with derivative in \({\cal{P}}\). Proc. Am. Math. Soc. 87(2), 251–257 (1983)
Ma, W. C., Minda, D.: A unified treatment of some special classes of functions. In: Proceedings of the Conference on Complex Analysis, Tianjin, 1992, pp. 157–169, Conf. Proc. Lecture Notes Anal. 1, Int. Press, Cambridge, MA, (1994)
Netanyahu, E.: The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in \( \left|z\right|<1\). Arch. Ration. Mech. Anal. 32, 100–112 (1969)
Noonan, J.W., Thomas, D.K.: On the second Hankel determinant of a really mean p-valent functions. Trans. Am. Math. Soc. 223, 337–346 (1976)
Lewin, M.: On a coefficient problem for bi-univalent functions. Proc. Am. Math. Soc. 18, 63–68 (1967)
Orhan, H., Magesh, N., Yamini, J.: Bounds for the second Hankel determinant of certain bi-univalent functions. Turkish J. Math. 40, 665–678 (2016)
Srivastava, H.M., Mishra, A.K., Das, M.K.: The Fekete-Szegö-problem for a subclass of close-to-convex functions. Complex Var. Elliptic Equ. 44(2), 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., Owa, S.: Current Topics in Analytic Function Theory. World Scientific, Singapore (1992)
Taha, T.S.: Topics in Uivalent Function Theory. Ph.D, University of London (1981)
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)
Zaprawa, P.: On the Fekete-Szegö problem for classes of bi-univalent functions. Bull. Belg. Math. Soc. Simon Stevin 1, 169–178 (2014)
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Mustafa, N., Mrugusundaramoorthy, G. & Janani, T. Second Hankel Determinant for a Certain Subclass of Bi-univalent Functions. Mediterr. J. Math. 15, 119 (2018). https://doi.org/10.1007/s00009-018-1165-1
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DOI: https://doi.org/10.1007/s00009-018-1165-1