• Orhan, Halit (Department of Mathematics Faculty of Science Ataturk University) ;
  • Zaprawa, Pawel (Department of Mathematics Lublin University of Technology)
  • Received : 2016.11.14
  • Accepted : 2017.05.23
  • Published : 2018.01.31


In this paper we obtain the bounds of the third Hankel determinants for the classes $\mathcal{S}^*({\alpha})$ of starlike functions of order ${\alpha}$ and $\mathcal{K}({\alpha}$) of convex functions of order ${\alpha}$. Moreover,we derive the sharp bounds for functions in these classes which are additionally 2-fold or 3-fold symmetric.


  1. Y. Abu Muhanna, L. Li, and S. Ponnusamy, Extremal problems on the class of convex functions of order -1/2, Arch. Math. 103 (2014), no. 6, 461-471.
  2. K. O. Babalola, On $H_3$(1) Hankel determinants for some classes of univalent functions, In: S. S. Dragomir and J. Y. Cho, editors. Inequality Theory and Applications. Nova Science Publishers New York, Vol. 6, 1-7, 2010.
  3. D. Bansal, S. Maharana, and J. K. Prajapat, Third order Hankel determinant for certain univalent functions, J. Korean Math. Soc. 52 (2015), no. 6, 1139-1148.
  4. D. Bshouty and A. Lyzzaik, Close-to-convexity criteria for planar harmonic mappings, Complex Anal. Oper. Theory 5 (2011), no. 3, 767-774.
  5. R. F. Gabriel, The Schwarzian derivative and convex functions, Proc. Amer. Math. Soc. 6 (1955), 58-66.
  6. T. Hayami and S. Owa, Generalized Hankel Determinant for Certain Classes, Int. J. Math. Anal. 4 (2010), no. 52, 2573-2585.
  7. W. K. Hayman, On the second Hankel determinant of mean univalent functions, Proc. Lond. Math. Soc. 18 (1968), 77-94.
  8. A. Janteng, S. A. Halim, and M. Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal. 1 (2007), no. 13, 619-625.
  9. A. Marx, Untersuchungen uber schlichte Abbildungen, Math. Ann. 107 (1933), no. 1, 40-67.
  10. J. W. Noonan and D. K. Thomas, On the Hankel determinants of areally mean p-valent functions, Proc. Lond. Math. Soc. 25 (1972), 503-524.
  11. K. I. Noor, On the Hankel determinant problem for strongly close-to-convex functions, J. Nat. Geom. 11 (1997), no. 1, 29-34.
  12. C. Pommerenke, On the coecients and Hankel determinants of univalent functions, J. Lond. Math. Soc. 41 (1966), 111-122.
  13. C. Pommerenke, On the Hankel determinants of univalent functions, Mathematika 14 (1967), 108-112.
  14. C. Pommerenke, Univalent functions, Vandenboeck and Ruprecht, Gottingen, 1975.
  15. S. Ponnusamy, S. K. Sahoo, and H. Yanagihara, Radius of convexity of partial sums of functions in the close-to-convex family, Nonlinear Anal. 95 (2014), 219-228.
  16. M. Raza and S. N. Malik, Upper bound of third Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli, J. Inequal. Appl. 2013 (2013), Art. 412, 8 pp.
  17. E. Strohhacker, Beitrage zur Theorie der schlichten Funktionen, Math. Z. 37 (1933), no. 1, 356-380.
  18. T. Umezawa, Analytic functions convex in one direction, J. Math. Soc. Japan 4 (1952), 194-202.
  19. D. Vamshee Krishna, B. Venkateswarlua, and T. RamReddy, Third Hankel determinant for bounded turning functions of order alpha, J. Nigerian Math. Soc. 34 (2015), no. 2, 121-127.
  20. D. Vamshee Krishna, B. Venkateswarlua, and T. RamReddy, Third Hankel determinant for certain subclass of p-valent functions, Complex Var. Elliptic Equ., doi:10.1080/17476933.2015.1012162, 2015.
  21. P. Zaprawa, Third Hankel determinant for classes of univalent functions, Mediterr. J. Math. 14 (2017), no. 1, Art. 19, 10 pp.