The sharp bounds of the second and third Hankel determinants for the class 𝓢𝓛*

Shagun Banga 1  and S. Sivaprasad Kumar 1
  • 1 Department of Applied Mathematics, Delhi Technological University, 110042, Delhi, India
Shagun Banga and S. Sivaprasad Kumar
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  • Department of Applied Mathematics, Delhi Technological University, Delhi, 110042, India
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Abstract

In this paper, we use the novel idea of incorporating the recently derived formula for the fourth coefficient of Carathéodory functions, in place of the routine triangle inequality to achieve the sharp bounds of the Hankel determinants H3(1) and H2(3) for the well known class 𝓢𝓛* of starlike functions associated with the right lemniscate of Bernoulli. Apart from that the sharp bound of the Zalcman functional: |a32a5| for the class 𝓢𝓛* is also estimated. Further, a couple of interesting results of 𝓢𝓛* are also discussed.

  • [1]

    Ali, R. M.—Cho, N. E.—Ravichandran, V.—Kumar, S. S.: Differential subordination for functions associated with the lemniscate of Bernoulli, Taiwanese J. Math. 16(3) (2012), 1017–1026.

    • Crossref
    • Export Citation
  • [2]

    Ali, R. M.—Jain, N. K.—Ravichandran, V.: Radii of starlikeness associated with the lemniscate of Bernoulli and the left-half plane, Appl. Math. Comput. 218(11) (2012), 6557–6565.

  • [3]

    Babalola, K. O.: On H 3(1) Hankel determinant for some classes of univalent functions, Inequal. Theory Appl. 6 (2010), 1–7.

  • [4]

    Bansal, D.: Upper bound of second Hankel determinant for a new class of analytic functions, Appl. Math. Lett. 26(1) (2013), 103–107.

    • Crossref
    • Export Citation
  • [5]

    Brown, J. E.—Tsao, A.: On the Zalcman conjecture for starlike and typically real functions, Math. Z. 191(3) (1986), 467–474.

    • Crossref
    • Export Citation
  • [6]

    Dienes, P.: The Taylor Series: An Introduction to the Theory of Functions of a Complex Variable, Dover Publications, Inc., New York, 1957.

  • [7]

    Goodman, A. W.: Univalent Functions, Vols. 1–2, Mariner, Tampa, FL, 1983.

  • [8]

    Hayman, W. K.: On the second Hankel determinant of mean univalent functions, Proc. London Math. Soc. 18(3) (1968), 77–94.

  • [9]

    Janteng, A.—Halim, S. A.—Darus, M.: Hankel determinant for starlike and convex functions, Int. J. Math. Anal. (Ruse) 1(13–16) (2007), 619–625.

  • [10]

    Kowalczyk, B.—Lecko, A.—Lecko, M.—Sim, Y. J.: The sharp bound of the third Hankel determinant for some classes of analytic functions, Bull. Korean Math. Soc. 55(6) (2018), 1859–1868.

  • [11]

    Kowalczyk, B.—Lecko, A.—Sim, Y. J.: The sharp bound for the Hankel determinant of the third kind for convex functions, Bull. Aust. Math. Soc. 97(3) (2018), 435–445.

    • Crossref
    • Export Citation
  • [12]

    Krishna, D. V.—Venkateswarlu, B.—Ramreddy, T.: Third Hankel determinant for bounded turning functions of order alpha, J. Nigerian Math. Soc. 34(2) (2015), 121–127.

    • Crossref
    • Export Citation
  • [13]

    Kumar, S. S.—Kumar, V.—Ravichandran, V.—Cho, N. E.: Sufficient conditions for starlike functions associated with the lemniscate of Bernoulli, J. Inequal. Appl. 2013, 2013:176, 13 pp.

  • [14]

    Kwon, O. S.—Lecko, A.—Sim, Y. J.: On the fourth coefficient of functions in the Carathéodory class, Comput. Methods Funct. Theory 18(2) (2018), 307–314.

    • Crossref
    • Export Citation
  • [15]

    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(2) (2019), 767–780.

    • Crossref
    • Export Citation
  • [16]

    Lee, S. K.—Ravichandran, V.—Supramaniam, S.: Bounds for the second Hankel determinant of certain univalent functions, J. Inequal. Appl. 2013, 2013:281, 17 pp.

  • [17]

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

    • Crossref
    • Export Citation
  • [18]

    Noor, K. I.: Hankel determinant problem for the class of functions with bounded boundary rotation, Rev. Roumaine Math. Pures Appl. 28(8) (1983), 731–739.

  • [19]

    Pommerenke, C.: On the coefficients and Hankel determinants of univalent functions, J. London Math. Soc. 41 (1966), 111–122.

  • [20]

    Ravichandran, V.—Verma, S.: Bound for the fifth coefficient of certain starlike functions, C. R. Math. Acad. Sci. Paris 353(6) (2015), 505–510.

    • Crossref
    • Export Citation
  • [21]

    Raza, M.—Malik, S. N.: Upper bound of the third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli, J. Inequal. Appl. 2013, 2013:412, 8 pp.

  • [22]

    Sokół, J.: Coefficient estimates in a class of strongly starlike functions, Kyungpook Math. J. 49(2) (2009), 349–353.

    • Crossref
    • Export Citation
  • [23]

    Sokół, J.: Radius problems in the class 𝒮ℒ*, Appl. Math. Comput. 214(2) (2009), 569–573.

  • [24]

    Sokół, J.—Stankiewicz, J.: Radius of convexity of some subclasses of strongly starlike functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat. 19 (1996), 101–105.

  • [25]

    Zaprawa, P.: Second Hankel determinants for the class of typically real functions, Abstr. Appl. Anal. 2016, Art. ID 3792367, 7 pp.

  • [26]

    Zaprawa, P.: Third Hankel determinants for subclasses of univalent functions, Mediterr. J. Math. 14(1) (2017), Art. 19, 10 pp.

  • [27]

    Zaprawa, P.: On Hankel determinant H 2(3) for univalent functions, Results Math. 73(3) (2018), Art. 89, 12 pp.

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