On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings

Year 2019, , 1695 - 1705, 08.12.2019

Yong Sun Zhi-gang Wang Antti Rasila

https://doi.org/10.15672/HJMS.2018.632

Cited By: 1

Abstract

In this paper, we obtain the upper bounds to the third Hankel determinants for convex functions of order $\alpha$ and bounded turning functions of order $\alpha$. Furthermore, several relevant results on a new subclass of close-to-convex harmonic mappings are obtained. Connections of the results presented here to those that can be found in the literature are also discussed.

Keywords

Univalent function, starlike function, convex function, bounded turning function, close-to-convex function, harmonic mapping, Hankel determinant

References

• [1] Y. Abu Muhanna, L. Li, and S. Ponnusamy, Extremal problems on the class of convex functions of order -1/2, Arch. Math. (Basel) 103 (6), 461–471, 2014.
• [2] K.O. Babalola, On H3(1) Hankel determinant for some classes of univalent functions, in: Inequal. Theory and Appl. 6, 1–7, editors: Y. J. Cho, J.K. Kim and S.S. Dragomir, 2010.
• [3] D. Bansal, Upper bound of second Hankel determinant for a new class of analytic functions, Appl. Math. Lett. 26 (1), 103–107, 2013.
• [4] D. Bansal, S. Haharana, and J.K. Prajapat, Third order Hankel determinant for certain univalent functions, J. Korean Math. Soc. 52, 1139–1148, 2015.
• [5] S.V. Bharanedhar and S. Ponnusamy, Coefficient conditions for harmonic univelent mappings and hypergeometric mappings, Rocky Mt. J. Math. 44, 753–777, 2014.
• [6] D. Bshouty, S.S. Joshi, and S.B. Joshi, On close-to-convex harmonic mappings, Complex Var. Elliptic Equ. 58, 1195–1199, 2013.
• [7] D. Bshouty and A. Lyzzaik, Close-to-convexity criteria for planar harmonic mappings, Complex Appl. Oper. Theory 5, 767–774, 2011.
• [8] D.G. Cantor, Power series with integral coefficients, Bull. Amer. Math. Soc. 69, 362– 366, 1963.
• [9] J. Chen, A. Rasila, and X. Wang, Coefficient estimates and radii problems for certain classes of polyharmonic mappings, Complex Var. Elliptic Equ. 60, 354–371, 2015.
• [10] J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A. I Math. 9, 3–25, 1984.
• [11] P. Dienes, The Taylor Series, Dover, New York, 1957.
• [12] P.L. Duren, Univalent functions, Springer Verlag, New York Inc., 1983.
• [13] P.L. Duren, Harmonic mappings in the plane, Cambridge University Press, Cambridge, 2004.
• [14] A. Edrei, Sur les déterminants récurrents et les singularités d’une fonction donnée par son développement de Taylor, Compos. Math. 7, 20–88, 1940.
• [15] M. Fekete and G. Szegő, Eine bemerkung über ungerade schlichte functions, J. London Math. Soc. 8, 85–89, 1933.
• [16] T. Hayami and S. Owa, Generalized Hankel determinant for certain classes, Int. J. Math. Anal. 4 (52), 2573–2585, 2010.
• [17] W.K. Hayman, On the second Hankel determinant of mean univalent functions, Proc. Lond. Math. Soc. 18 (3), 77–94, 1968.
• [18] A. Janteng, S. Halim, and M. Darus, Coefficient inequality for a function whose derivative has a positive real part, J. Inequal. Pure Appl. Math. 7 (2), Art. 50, 5 pp, 2006.
• [19] A. Janteng, S.A. Halim, and M. Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal. 1 (13), 619–625, 2007.
• [20] D. Kalaj, S. Ponnusamy, and M. Vuorinen, Radius of close-to-convexity and fully starlikeness of harmonic mappings, Complex Var. Elliptic Equ. 59, 539–552, 2014.
• [21] W. Koepf, On the Fekete-Szegö problem for close-to-convex functions, Proc. Amer. Math. Soc. 101, 89–95, 1987.
• [22] W. Koepf, On the Fekete-Szegö problem for close-to-convex functions II, Arch. Math. 49, 420–433, 1987.
• [23] S.K. Lee, V. Ravichandran, and S. Subramaniam, Bounds for the second Hankel determinant of certain univalent functions, J. Inequal. Appl. 2013, Art. 281, 17 pp, 2013.
• [24] R.J. Libera and E.J. Złotkiewicz, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85, 225–230, 1982.
• [25] R.J. Libera and E.J. Złotkiewicz, Coefficient bounds for the inverse of a function with derivatives in P, Proc. Am. Math. Soc. 87, 251–257, 1983.
• [26] A.E. Livingston, The coefficients of multivalent close-to-convex functions, Proc. Am. Math. Soc. 21, 545–552, 1969.
• [27] P.T. Mocanu, Sufficient conditions of univalence for complex functions in the class C1, Rev. Anal. Numér. Théor. Approx. 10, 75–79, 1981.
• [28] P.T. Mocanu, Injectivity conditions in the complex plane, Complex Appl. Oper. Theory 5, 759–766, 2011.
• [29] S. Nagpal and V. Ravichandran, A subclass of close-to-convex harmonic mappings, Complex Var. Elliptic Equ. 59, 204–216, 2014.
• [30] H. Orhan and P. Zaprawa, Third Hankel determinants for starlike and convex functions, Bull. Korean Math. Soc. 55, 165–173, 2018.
• [31] C. Pommerenke, On the coefficients and Hankel determinant of univalent functions, J. Lond. Math. Soc. 41, 111–122, 1966.
• [32] C. Pommerenke, On the Hankel determinants of univalent functions, Mathematika 14, 108–112, 1967.
• [33] S. Ponnusamy and A. Rasila, Planar harmonic and quasiregular mappings, in: Topics in modern function theory, Ramanujan Math. Soc. Lect. Notes Ser. 19, 267–333, Ramanujan Math. Soc., Mysore, 2013.
• [34] S. Ponnusamy and A. Sairam Kaliraj, On harmonic close-to-convex functions, Comput. Methods Funct. Theory 12, 669–685, 2012.
• [35] S. Ponnusamy and A. Sairam Kaliraj, Constants and characterization for certain classes of univalent harmonic mappings, Mediterr. J. Math. 12, 647–665, 2015.
• [36] M. Raza and S.N. Malik, Upper bound of the third Hankel determinant for a class of analytic functions related with Lemniscate of Bernoulli, J. Inequal. Appl. 2013, Art. 412, 8 pp., 2013.
• [37] T.J. Suffridge, Some special classes of conformal mappings, in: Handbook of complex analysis: geometric function theory (Edited by Kühnau), 2, 309–338, Elsevier, Amsterdam, 2005.
• [38] Y. Sun, Y. Jiang, and A. Rasila, On a subclass of close-to-convex harmonic mappings, Complex Var. Elliptic Equ. 61, 1627–1643, 2016.
• [39] D. Vamshee Krishna, B. Venkateswarlu, and T. RamReddy, Third Hankel determinant for bounded turning functions of order alpha, J. Nigerian Math. Soc. 34, 121–127, 2015.
• [40] P. Zaprawa, Third Hankel determinants for subclasses of univalent functions, Mediterr. J. Math. 14 (1), Art. 19, 10 pp., 2017.