Hankel determinant for a class of analytic functions of complex order defined by convolution

S. M. El-Deeb, M. K. Aouf

Abstract


In this paper, we obtain the Fekete-Szego inequalities for the functions of complex order defined by convolution. Also, we find upper bounds for the second Hankel determinant \(|a_2a_4-a_3^2|\) for functions belonging to the class \(S_{\gamma}^b(g(z);A,B)\).

Keywords


Fekete-Szego inequality; second Hankel determinant; convolution; complex order.

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References


Abubaker, A., Darus, M., Hankel determinant for a class of analytic functions involving a generalized linear differential operator, Internat. J. Pure Appl. Math. 69 (3) (2011), 429-435.

Al-Oboudi, F. M., On univalent functions defined by a generalized Salagean operator, Internat. J. Math. Math. Sci. 27 (2004), 1429-1436.

Aouf, M. K., Subordination properties for a certain class of analytic functions defined by the Salagean operator, Appl. Math. Lett. 22 (2009), 1581-1585.

Bansal, D., Upper bound of second Hankel determinant for a new class of analytic functions}, Appl. Math. Lett. 26 (1) (2013), 103-107.

Bansal, D., Fekete-Szego problem and upper bound of second Hankel determinant for a new class of analytic functions, Kyungpook Math. J. 54 (2014), 443-452.

Bulboaca, T., Differential Subordinations and Superordinations, Recent Results, House of Scientific Book Publ., Cluj-Napoca, 2005.

Catas, A., On certain classes of p-valent functions defined by multiplier transformations, in Proceedings of the International Symposium on Geometric Function Theory and Applications, (Istanbul, Turkey, August 2007), Istanbul, 2008, 241-250.

Cho, N. E., Srivastava, H. M., Argument estimates of certain analytic functions defined by a class of multiplier transformations, Math. Comput. Modelling 37 (1-2) (2003), 39-49.

Cho, N. E., Kim, T. H., Multiplier transformations and strongly close-to-convex functions, Bull. Korean Math. Soc. 40 (3) (2003), 399-410.

Duren, P. L., Univalent Functions, Springer-Verlag, New York, 1983.

Dziok, J., Srivastava, H. M., Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput. 103 (1999), 1-13.

Dziok, J., Srivastava, H. M., Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transforms Spec. Funct. 14 (2003), 7-18.

El-Ashwah, R. M., Aouf, M. K., Differential subordination and superordination for certain subclasses of p-valent functions, Math. Comput. Modelling 51 (2010), 349-360.

El-Ashwah, R. M., Aouf, M. K., Some properties of new integral operator, Acta Univ. Apulensis 24 (2010), 51-61.

Grenander, U., Szego, G., Toeplitz Forms and Their Application, Univ. of California Press, Berkeley, 1958.

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

Keogh, F. R., Markes, E. P., A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20 (1969), 8-12.

Libera, R. J., Złotkewicz, E. J., Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc. 87 (2) (1983), 251-257.

Miller, S. S., Mocanu, P. T., Differential Subordination: Theory and Applications, Marcel Dekker Inc., New York-Basel, 2000.

Mishra, A. K., Gochhayat, P., Second Hankel determinant for a class of analytic functions defined by fractional derivative, Int. J. Math. Math. Sci. (2008), Art. ID 153280, 1-10.

Mishra, A. K., Kund, S. N., Second Hankel determinant for a class of functions defined by the Carlson-Shaffer, Tamkang J. Math. 44 (1) (2013), 73-82.

Mohammed, A., Darus, M., Second Hankel determinant for a class of analytic functions defined by a linear operator, Tamkang J. Math. 43 (3) (2012), 455-462.

Noonan, J. W., Thomas, D. K., On the second Hankel determinant of areally mean p-valent functions, Trans. Amer. Math. Soc. 223 (2) (1976), 337-346.

Prajapat, J. K., Subordination and superordination preserving properties for generalized multiplier transformation operator, Math. Comput. Modelling 55 (2012), 1456-1465.

Raina, R. K., Bansal, D., Some properties of a new class of analytic functions defined in tems of a Hadmard product, J. Inequal. Pure Appl. Math. 9 (2008), Art. 22, 1-9.

Rogosinski, W., On the coefficients of subordinate functions, Proc. London Math. Soc. 48 (1943), 48-82.

Salagean, G. S., Subclasses of univalent functions, in Complex analysis - fifth Romanian-Finnish seminar, Part 1 (Bucharest, 1981), Springer-Verlag, Berlin, 1983, 362-372.

Selvaraj, C., Karthikeyan, K. R., Differential subordinant and superordinations for certain subclasses of analytic functions, Far East J. Math. Sci. 29 (2) (2008), 419-430.

Srivastava, H. M., Karlsson, P. W., Multiple Gaussian Hypergeometric Series, Ellis Horwood Limited, Chichester; Halsted Press, New York, 1985.




DOI: http://dx.doi.org/10.17951/a.2015.69.2.47-59
Date of publication: 2015-12-30 22:51:59
Date of submission: 2015-12-30 12:54:27


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