×

Upper bound of second Hankel determinant for certain subclasses of analytic functions. (English) Zbl 1474.30088

Summary: In this present investigation, we first give a survey of the work done so far in this area of Hankel determinant for univalent functions. Then the upper bounds of the second Hankel determinant \(| a_2 a_4 - a_3^2 |\) for functions belonging to the subclasses \(S(\alpha, \beta)\), \(K(\alpha, \beta)\), \(S_s^\ast(\alpha, \beta)\), and \(K_s(\alpha, \beta)\) of analytic functions are studied. Some of the results, presented in this paper, would extend the corresponding results of earlier authors.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Graham, I.; Kohr, G., Geometric Function Theory in One and Higher Dimensions (2003), New York, NY, USA: Marcel Dekker, New York, NY, USA · Zbl 1042.30001 · doi:10.1201/9780203911624
[2] Sakaguchi, K., On a certain univalent mapping, Journal of the Mathematical Society of Japan, 11, 72-75 (1959) · Zbl 0085.29602 · doi:10.2969/jmsj/01110072
[3] Das, R. N.; Singh, P., On subclasses of schlicht mapping, Indian Journal of Pure and Applied Mathematics, 8, 8, 864-872 (1977) · Zbl 0374.30008
[4] Wang, Z. G.; Jiang, Y. P., On certain subclasses of close-to-convex and quasi-convex functions with respect to \(2 k\)-symmetric conjugate points, Journal of Mathematics and Applications, 29, 167-179 (2007) · Zbl 1155.30341
[5] Wang, Z. G.; Gao, C. Y.; Liu, M. S.; Liao, M. X., On subclasses of close-to-convex and quasi-convex functions with respect to \(k\)-symmetric points, Advances in Mathematics, 38, 1, 44-56 (2009) · Zbl 1482.30053
[6] Sudharsan, T. V.; Balasubrahmanyam, P.; Subramanian, K. G., On functions starlike with respect to symmetric and conjugate points, Taiwanese Journal of Mathematics, 2, 1, 57-68 (1998) · Zbl 0909.30009
[7] Pommerenke, C., On the coefficients and Hankel determinants of univalent functions, Journal of the London Mathematical Society, 41, 111-122 (1966) · Zbl 0138.29801
[8] Noonan, J. W.; Thomas, D. K., On the Hankel determinants of areally mean \(p\)-valent functions, Proceedings of the London Mathematical Society, 25, 503-524 (1972) · Zbl 0252.30019
[9] Pommerenke, C., On the Hankel determinants of univalent functions, Mathematika, 14, 108-112 (1967) · Zbl 0165.09602
[10] Noor, K. I., On the Hankel determinants of close-to-convex univalent functions, International Journal of Mathematics and Mathematical Sciences, 3, 3, 447-481 (1980) · Zbl 0443.30015 · doi:10.1155/S016117128000035X
[11] Noor, K. I., On certain analytic functions related with strongly close-to-convex functions, Applied Mathematics and Computation, 197, 1, 149-157 (2008) · Zbl 1133.30308 · doi:10.1016/j.amc.2007.07.039
[12] Noor, K. I., Hankel determinant problem for the class of functions with bounded boundary rotation, Revue Roumaine de Mathématiques Pures et Appliquées, 28, 8, 731-739 (1983) · Zbl 0524.30008
[13] UI-Haq, W.; Noor, K. I., A certain class of analytic functions and the growth rate of Hankel feterminant, Journal of Inequalities and Applications, 2012, article 309 (2012) · Zbl 1296.30028
[14] Arif, M.; Noor, K. I.; Raza, M., Hankel determinant problem of a subclass of analytic functions, Journal of Inequalities and Applications, 2012, 22-27 (2012) · Zbl 1296.30014 · doi:10.1186/1029-242X-2012-22
[15] Arif, M.; Noor, K. I.; Raza, M.; Haq, W., Some properties of a generalized class of analytic functions related with Janowski functions, Abstract and Applied Analysis, 2012 (2012) · Zbl 1242.30009 · doi:10.1155/2012/279843
[16] Ehrenborg, R., The Hankel determinant of exponential polynomials, The American Mathematical Monthly, 107, 6, 557-560 (2000) · Zbl 0985.15006 · doi:10.2307/2589352
[17] Layman, J. W., The Hankel transform and some of its properties, Journal of Integer Sequences, 4, 1, 1-11 (2001) · Zbl 0978.15022
[18] Hayman, W. K., On the second Hankel determinant of mean univalent functions, Proceedings of the London Mathematical Society, 18, 3, 77-94 (1968) · Zbl 0158.32101
[19] Ali, R. M.; Lee, S. K.; Ravichandran, V.; Supramaniam, S., The Fekete-Szegö coefficient functional for transforms of analytic functions, Bulletin of the Iranian Mathematical Society, 35, 2, 119-142 (2009) · Zbl 1193.30006
[20] Ali, R. M.; Ravichandran, V.; Supramaniam, S., Coefficient bounds for \(p\)-valent functions, Applied Mathematics and Computation, 187, 1, 35-46 (2007) · Zbl 1113.30024 · doi:10.1016/j.amc.2006.08.100
[21] Hayami, T.; Owa, S., Coefficient estimates for analytic functions concerned with Hankel determinant, Fractional Calculus & Applied Analysis, 13, 4, 367-384 (2010) · Zbl 1221.30031
[22] Hayami, T.; Owa, S., Generalized Hankel determinant for certain classes, International Journal of Mathematical Analysis, 4, 52, 2573-2585 (2010) · Zbl 1226.30015
[23] Krishna, D. V.; Ramreddy, T., Hankel determinant for \(p\)-valent starlike and convex functions of order \(\alpha \), Novi Sad Journal of Mathematics, 42, 2, 89-102 (2012) · Zbl 1289.30123
[24] Hayami, T.; Owa, S., Hankel determinant for \(p\)-valently starlike and convex functions of order \(\alpha \), General Mathematics, 17, 4, 29-44 (2009) · Zbl 1289.30071
[25] Hayami, T.; Owa, S., Applications of Hankel determinant for \(p\)-valently starlike and convex functions of order \(\alpha \), Far East Journal of Applied Mathematics, 46, 1, 1-23 (2010) · Zbl 1284.30007
[26] Janteng, A.; Halim, S. A.; Darus, M., Coefficient inequality for a function whose derivative has a positive real part, Journal of Inequalities in Pure and Applied Mathematics, 7, 2, 1-5 (2006) · Zbl 1134.30310
[27] Janteng, A.; Halim, S. A.; Darus, M., Hankel determinant for starlike and convex functions, International Journal of Mathematical Analysis, 1, 13, 619-625 (2007) · Zbl 1137.30308
[28] Janteng, A.; Halim, S. A.; Darus, M., Hankel determinant for functions starlike and convex with respect to symmetric points, Journal of Quality Measurement and Analysis, 2, 1, 37-43 (2006)
[29] Singh, G., Hankel determinant for analytic functions with respect to other points, Engineering Mathematics Letters, 2, 2, 115-123 (2013)
[30] Mishra, A. K.; Gochhayat, P., Second Hankel determinant for a class of analytic functions defined by fractional derivative, International Journal of Mathematics and Mathematical Sciences, 2008 (2008) · Zbl 1158.30308 · doi:10.1155/2008/153280
[31] Owa, S.; Srivastava, H. M., Univalent and starlike generalized hypergeometric functions, Canadian Journal of Mathematics, 39, 5, 1057-1077 (1987) · Zbl 0611.33007 · doi:10.4153/CJM-1987-054-3
[32] Mohammed, A.; Darus, M., Second Hankel determinant for a class of analytic functions defined by a linear operator, Tamkang Journal of Mathematics, 43, 3, 455-462 (2012) · Zbl 1255.30019 · doi:10.5556/j.tkjm.43.2012.455-462
[33] Ibrahim, W. R., Bounded nonlinear functional derived by the generalized Srivastava-Owa fractional differential operator, International Journal of Analysis, 2013 (2013) · Zbl 1268.34016
[34] Mishra, A. K.; Kund, S. N., The second Hankel determinant for a class of analytic functions associated with the Carlson-Shaffer operator, Tamkang Journal of Mathematics, 44, 1, 73-82 (2013) · Zbl 1278.30016
[35] Singh, G., Hankel determinant for a new subclass of analytic functions, Scientia Magna, 8, 4, 61-65 (2012)
[36] Mehrok, B. S.; Singh, G., Estimate of second Hankel determinant for certain classes of analytic functions, Scientia Magna, 8, 3, 85-94 (2012)
[37] Shanmugam, G.; Adolf Stephen, B.; Subramanian, K. G., Second Hankel determinant for certain classes of analytic functions, Bonfring International Journal of Data Mining, 2, 2, 57-60 (2012)
[38] Krishna, D. V.; Ramreddy, T., Coefficient inequality for certain subclass of analytic functions, Armenian Journal of Mathematics, 4, 2, 98-105 (2012) · Zbl 1281.30017
[39] Abubaker, A.; Darus, M., Hankel determinant for a class of analytic functions involving a generalized linear differential operator, International Journal of Pure and Applied Mathematics, 69, 4, 429-435 (2011) · Zbl 1220.30011
[40] Al-Abbadi, M. H.; Darus, M., Hankel determinant for certain class of analytic function defined by geberalized derivative operator, Tamkang Journal of Mathematics, 43, 3, 445-453 (2012) · Zbl 1259.30010 · doi:10.5556/j.tkjm.43.2012.445-453
[41] Bansal, D., Upper bound of second Hankel determinant for a new class of analytic functions, Applied Mathematics Letters, 26, 1, 103-107 (2013) · Zbl 1250.30006 · doi:10.1016/j.aml.2012.04.002
[42] Deekonda, V. K.; Thoutreddy, R., An upper bound to the second Hankel determinant for a subclass of analytic functions, Bulletin of International Mathematical Virtual Institute, 4, 1, 17-26 (2014) · Zbl 1446.30023
[43] Krishna, D. V.; Ramreddy, T., Coefficient inequality for certain subclasses of analytic functions, New Zealand Journal of Mathematics, 42, 217-228 (2012) · Zbl 1279.30028 · doi:10.1080/03036758.2011.559664
[44] Lee, S. K.; Ravichandran, V.; Supramaniam, S., Bounds for the second Hankel determinant of certain univalent functions, Journal of Inequalities and Applications, 2013, article 281 (2013) · Zbl 1302.30018 · doi:10.1186/1029-242X-2013-281
[45] Murugusundaramoorthy, G.; Magesh, N., Coefficient inequalities for certain classes of analytic functions associated with Hankel determinant, Bulletin of Mathematical Analysis and Applications, 1, 3, 85-89 (2009) · Zbl 1312.30024
[46] Mohamed, N.; Mohamad, D.; Soh, S. C., Second Hankel determinant for certain generalized classes of analytic functions, International Journal of Mathematical Analysis, 6, 17, 807-812 (2012) · Zbl 1247.30023
[47] Noonan, J. W.; Thomas, D. K., On the second Hankel determinant of areally mean \(p\)-valent functions, Transactions of the American Mathematical Society, 223, 2, 337-346 (1976) · Zbl 0346.30012
[48] Verma, S.; Gupta, S.; Singh, S., Bounds of Hankel determinant for a class of univalent functions, International Journal of Mathematics and Mathematical Sciences, 2012 (2012) · Zbl 1256.30012
[49] Yahya, A.; Soh, S. C.; Mohamad, D., Second Hankel determinant for a class of a generalised Sakaguchi class of analytic functions, International Journal of Mathematical Analysis, 7, 33, 1601-1608 (2013) · Zbl 1283.30045
[50] Raza, M.; Malik, S. N., Upper bound of third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli, Journal of Inequalities and Applications, 2013, article 412 (2013) · Zbl 1291.30106
[51] Babalola, K. O., On \(H_3(1)\) Hankel determinants for some classes of univalent functions, Inequality Theory and Applications, 6, 1-7 (2010), Nova Science
[52] Duren, P. L., Univalent functions, Grundlehren der Mathematischen Wissenschaften, 259 (1983), New York, NY, USA: Springer, New York, NY, USA · Zbl 0514.30001
[53] Libera, R. J.; Złotkiewicz, E. J., Early coefficients of the inverse of a regular convex function, Proceedings of the American Mathematical Society, 85, 2, 225-230 (1982) · Zbl 0464.30019 · doi:10.2307/2044286
[54] Libera, R. J.; Złotkiewicz, E. J., Coefficient bounds for the inverse of a function with derivative in \(\mathcal{P} \), Proceedings of the American Mathematical Society, 87, 2, 251-257 (1983) · Zbl 0488.30010 · doi:10.2307/2043698
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.