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Investigation of the fifth Hankel determinant for a family of functions with bounded turnings

Muhammad Arif, Inayat Ullah, Mohsan Raza and Paweł Zaprawa
From the journal Mathematica Slovaca

Abstract

The main aim of this paper is to study the fifth Hankel determinant for the class of functions with bounded turnings. The results are also investigated for 2-fold symmetric and 4-fold symmetric functions.

MSC 2010: Primary 30C45; 30C50

  1. (Communicated by Stanisława Kanas )

References

[1] Altinkaya, Ş.—Yalçin, S.: Third Hankel determinant for Bazilevič functions, Adv. Math. 5 (2016), 91–96. Search in Google Scholar

[2] Altinkaya, Ş.—Yalçin, S.: Upper bound of second Hankel determinant for bi-Bazilevic functions, Mediterr. J. Math. 13 (2016), 4081–4090. Search in Google Scholar

[3] Arif, M.—Noor, K. I.—Raza, M.: Hankel determinant problem of a subclass of analytic functions, J. Inequal. Appl. 2012 (2012), #22. Search in Google Scholar

[4] Arif, M.—Noor, K. I.—Raza, M.—Haq, W.: Some properties of a generalized class of analytic functions related with Janowski functions, Abstr. Appl. Anal. 2012 (2012), Art. ID 279843. Search in Google Scholar

[5] Arif, M.—Rani, L.—Raza, M.—Zaprawa, P.: Fourth Hankel determinant for a family of functions with bounded turning, Bull. Korean Math. Soc. 55 (2018), 1703–1711. Search in Google Scholar

[6] Babalola, K. O.: On H3(1) Hankel determinant for some classes of univalent functions, Inequal. Theory Appl. 6 (2010), 1–7. Search in Google Scholar

[7] Bansal, D.: Upper bound of second Hankel determinant for a new class of analytic functions, Appl. Math. Lett. 26 (2013), 103–107. Search in Google Scholar

[8] Bansal, D.—Maharana, S.—Prajapat, J. K.: Third order Hankel Determinant for certain univalent functions, J. Korean Math. Soc. 52 (2015), 1139–1148. Search in Google Scholar

[9] Çaglar, M.—Deniz, E.—Srivastava, H. M.: Second Hankel determinant for certain subclasses of bi-univalent functions, Turk. J. Math. 41 (2017), 694–706. Search in Google Scholar

[10] Caratheodory, C.: über den Variabilitätsbereich der fourierschen Konstanten von positiven harmonischen Funktionen, Rend. Circ. Mat. Palermo 32 (1911), 193–217. Search in Google Scholar

[11] Hayman, W. K.: On second Hankel determinant of mean univalent functions, Proc. London Math. Soc. 3 (1968), 77–94. Search in Google Scholar

[12] 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), 1–5. Search in Google Scholar

[13] Janteng, A.—Halim, S. A.—Darus, M.: Hankel determinant for starlike and convex functions, Int. J. Math. Anal. 1 (2007), 619–625. Search in Google Scholar

[14] Kowalczyk, B.—Lecko, A.—Sim, Y. J.: The sharp bound of the Hankel determinant of the third kind for convex functions, Bull. Aust. Math. Soc. 97 (2018), 435–445. Search in Google Scholar

[15] Krishna, D. V.—Venkateswarlu, B.—RamReddy, T.: Third Hankel determinant for bounded turning functions of order alpha, J. Nigerian Math. Soc. 34 (2015), 121–127. Search in Google Scholar

[16] Kwon, O. S.—Lecko, A.—Sim, Y. J.: The bound of the Hankel determinant of the third kind for starlike functions, Bull. Malays. Math. Sci. Soc. 42 (2019), 767–780. Search in Google Scholar

[17] Lee, S. K.—Ravichandran, V.—Supramaniam, S.: Bounds for the second Hankel determinant of certain univalent functions, J. Inequal. Appl. 2013, (2013), #281. Search in Google Scholar

[18] Lecko, A.—Sim, Y. J.—Śmiarowska, B.: The sharp bound of the Hankel determinant of the third kind for starlike functions of order 1/2, Complex Anal. Oper. Theory 2018 (2018), 1–8. Search in Google Scholar

[19] Liu, M. S.—Xu, J. F.—Yang, M.: Upper bound of second Hankel determinant for certain subclasses of analytic functions, Abstr. Appl. Anal. 2014 (2014), Art. ID 603180. Search in Google Scholar

[20] Livingston, A. E.: The coefficients of multivalent close-to-convex functions, Proc. Amer. Math. Soc. 21 (1969), 545–552. Search in Google Scholar

[21] 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. Search in Google Scholar

[22] Noor, K. I.: Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roumaine Math. Pures Appl. 28 (1983), 731–739. Search in Google Scholar

[23] Noor, K. I.: On certain analytic functions related with strongly close-to-convex functions, Appl. Math. Comput. 197 (2008), 149–157. Search in Google Scholar

[24] Orhan, H.—Magesh, N.—Yamini, J.: Bounds for the second Hankel determinant of certain bi-univalent functions, Turk. J. Math. 40 (2016), 679–687. Search in Google Scholar

[25] Pommerenke, C.: On the coefficients and Hankel determinants of univalent functions, J. London Math. Soc. 1 (1966), 111–122. Search in Google Scholar

[26] Pommerenke, C.: On the Hankel determinants of univalent functions, Mathematika 14 (1967), 108–112. Search in Google Scholar

[27] Răducanu, D.—Zaprawa, P.: Second Hankel determinant for close-to-convex functions, C. R. Math. Acad. Sci. Paris 355 (2017), 1063–1071. Search in Google Scholar

[28] Raza, M.—Malik, S. N.: Upper bound of third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli, J. Inequal. Appl. 2013 (2013), #412. Search in Google Scholar

[29] Shanmugam, G.—Stephen, B. A.—Babalola, K. O.: Third Hankel determinant for α-starlike functions, Gulf J. Math. 2 (2014), 107–113. Search in Google Scholar

[30] Shi, L.—Ali, I.—Arif, M.—Cho, N. E.—Hussain, S.—Khan, H.: A study of third Hankel determinant problem for certain subfamilies of analytic functions involving cardioid domain, Mathematics 11 (2019), 15 pages. Search in Google Scholar

[31] Shi, L.—Srivastava, H. M.—Arif, M.—Hussain, S.—Khan H.: An investigation of the third Hankel determinant problem for certain subfamilies of univalent functions involving the exponential function, Symmetry 11 (2019), #598. Search in Google Scholar

[32] Zaprawa, P.: Third Hankel determinants for subclasses of univalent functions, Mediterr. J. Math. 14 (2017), #19. Search in Google Scholar

[33] Zhang, H-Y.—Tang, H.—Niu, X-M.: Third-order Hankel determinant for certain class of analytic functions related with exponential function, Symmetry 10 (2018), #501. Search in Google Scholar

Received: 2019-05-24
Accepted: 2019-09-15
Published Online: 2020-03-10
Published in Print: 2020-04-28

© 2020 Mathematical Institute Slovak Academy of Sciences