Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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THIRD HANKEL DETERMINANTS FOR STARLIKE AND CONVEX FUNCTIONS OF ORDER ALPHA

  • 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
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Abstract

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.

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References

  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. https://doi.org/10.4134/JKMS.2015.52.6.1139
  4. D. Bshouty and A. Lyzzaik, Close-to-convexity criteria for planar harmonic mappings, Complex Anal. Oper. Theory 5 (2011), no. 3, 767-774. https://doi.org/10.1007/s11785-010-0056-7
  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. https://doi.org/10.1007/BF01448878
  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. https://doi.org/10.1112/S002557930000807X
  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. https://doi.org/10.1016/j.na.2013.09.009
  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. https://doi.org/10.1186/1029-242X-2013-8
  17. E. Strohhacker, Beitrage zur Theorie der schlichten Funktionen, Math. Z. 37 (1933), no. 1, 356-380. https://doi.org/10.1007/BF01474580
  18. T. Umezawa, Analytic functions convex in one direction, J. Math. Soc. Japan 4 (1952), 194-202. https://doi.org/10.2969/jmsj/00420194
  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. https://doi.org/10.1016/j.jnnms.2015.03.001
  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. https://doi.org/10.1007/s00009-016-0817-2