On certain subclasses of analytic functions associated with the Carlson–Shaffer operator

Jagannath Patel, Ashok Kumar Sahoo

Abstract


The object of the present paper is to solve Fekete-Szego problem and determine the sharp upper bound to the second Hankel determinant for a certain class \(R^{\lambda}(a,c,A,B)\) of analytic functions in the unit disk. We also investigate several majorization properties for functions belonging to a subclass \(\widetilde {R}^{\lambda}(a,c, A,B)\) of \(R^{\lambda}(a,c,A,B)\) and related function classes. Relevant connections of the main results obtained here with those given by earlier workers on the subject are pointed out.

Full Text:

PDF

References


Altinta¸s, O., Özkan, O., Srivastava, H. M., Majorization by starlike functions of complex order, Complex Var. 46 (2001), 207–218.

Caplinger, T. R., Causey, W. M., A class of univalent functions, Proc. Amer. Math. Soc. 39 (1973), 357–361.

Carlson, B. C., Shaffer, D. B., Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal. 15 (1984), 737–745.

Duren, P. L., Univalent Functions, Grundlehren der MathematischenWissenschaften, 259, Springer-Verlag, New York, USA, 1983.

Fekete, M., Szegö, G., Eine Bemerkung über ungerede schlichte funktionen, J. London Math. Soc. 8 (1933), 85–89.

Goyal, S. P., Goswami, P., Majorization for certain subclass of analytic functions defined by linear operator using differential subordination, Appl. Math. Letters 22 (2009), 1855–1858.

Goyal, S. P., Bansal, S. K., Goswami, P., Majorization for certain classes of functions by fractional derivatives, J. Appl. Math. Stat. Informatics 6(2) (2010), 45–50.

Janteng, A., Halim, S. A., Darus, M., Coefficient inequality for a function whose derivative has a positive real part, J. Inequal. Pure Appl. Math. 7 (2006), Art. 50 [http://jipam.vu.edu.au/].

Janteng, A., Halim, S. A., Darus, M., Estimate on the second Hankel functional for functions whose derivative has a positive real part, J. Quality Measurement and Analysis 4 (2008), 189–195.

Juneja, O. P., Mogra, M. L., A class of univalent functions, Bull. Sci. Math. (2) 103 (1979), 435–447.

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

Koepf, W., On the Fekete-Szegö problem for close-to-convex functions. II, Arch. Math. (Basel) 49 (1987), 420–433.

Koepf, W., On the Fekete-Szegö problem for close-to-convex functions, Proc. Amer. Math. Soc. 101 (1987), 89–95.

Libera, R. J., Złotkiewicz, E. J., Early coefficient of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85 (2) (1982), 225–230.

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

Ma, W. C., Minda, D., A unified treatment of some special classes of univalent functions, Proceedings of the Conference on Complex Analysis (Tianjin, 1992), Z. Li, F. Ren, L. Yang and S. Zhang (Eds.), Int. Press, Cambridge, MA, 1994, 157–169.

MacGregor, T. H., Functions whose derivative have a positive real part, Trans. Amer. Math. Soc. 104(3) (1962), 532–537.

MacGregor, T. H., The radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 14 (1963), 514–520.

MacGregor, T. H., Majorization by univalent functions, Duke Math. J. 34 (1967), 95–102.

Miller, S. S., Mocanu, P. T., Differential Subordinations: Theory and Applications, Series on Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, Marcel Dekker Inc., New York and Basel, 2000.

Mishra, A. K., Kund, S. N., The second Hankel determinant for a class of analytic functions associated with the Carlson-Shaffer operator, Tamkang J. Math. 44(1) (2013), 73–82.

Nehari, Z., Conformal Mapping, McGraw-Hill Book Company, New York, Toronto and London, 1952.

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

Noor, K. I., Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roum. Math. Pures Appl. 28 (1983), no. 8, 731–739.

Owa, S., Srivastava, H. M., Univalent and starlike generalized hypergeometric functions, Canad. J. Math. 39 (1987), 1057–1077.

Ruscheweyh, St., New criteria for univalent functions, Proc. Amer. Math. Soc. 49 (1975), 109–115.

Silverman, H., Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51 (1975), 109–116.

Srivastava, H. M., Karlson, P. W., Karlsson, Per W., Multiple Gaussian Hypergeometric Series (Mathematics and its Applications), A Halsted Press Book (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985.




DOI: http://dx.doi.org/10.17951/a.2014.68.2.65
Date of publication: 2015-05-23 16:29:45
Date of submission: 2015-05-09 13:48:57


Statistics


Total abstract view - 607
Downloads (from 2020-06-17) - PDF - 240

Indicators



Refbacks

  • There are currently no refbacks.


Copyright (c) 2015 Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica