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

Year 2019, Volume: 48 Issue: 6 , 1695 - 1705 , 08.12.2019

Yong Sun Zhi-gang Wang Antti Rasila

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

Cited By: 1

https://izlik.org/JA37UL26GB

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.