ON A SUBCLASS OF STARLIKE FUNCTIONS
ASSOCIATED WITH GENERALIZED CARDIOID

Abstract

The purpose of this paper is to investigate a subclass of analytic functions associated with generalized cardioid in the open unit disk. The geometric properties of functions in the subclass are investigated. Subsequently, the bound for initial coefficients, the Fekete-Szego inequality and second Hankel determinant inequality for functions belonging to this class are obtained. Furthermore, we find the sharp estimate for Toeplitz determinant, $T_2(2)$ for this class.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 33
Issue: 5
Year: 2020

DOI: 10.12732/ijam.v33i5.3

Download Section



Download the full text of article from here.

You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.

References

  1. [1] R.M. Ali, N.K. Jain and V. Ravichandran, On the radius constants for classes of analytic functions, Bull. Malays. Math. Sci.Soc, 36, No 1 (2013), 23-38.
  2. [2] R.M. Ali, N.K. Jain and V. Ravichandran, Radii of starlikeness associated with the lemniscate of Bernoulli and left-half plane, Applied Mathematics and Computation, 218, No 11 (2012), 6557-6565.
  3. [3] M.K. Aouf, J. Dziok and J. Sokol, On a subclass of strongly starlike functions, Applied Mathematics Letters, 24, No 1 (2011), 27-32.
  4. [4] P.L. Duren, Univalent Functions (Grundlehren der mathematischen Wissenschaften) 259), Springer-Verlag, New York-Berlin-Heidelberg-Tokyo (1983).
  5. [5] J. Dziok, R.K. Raina and J. Sokol, On a class of starlike functions related to a shell-like curve connected with Fibonacci numbers, Mathematical and Computer Modelling 57, (2013), 1203-1211.
  6. [6] I. Efraimidis, A generalization of Livingston’s coefficient inequalities for functions with positive real part, Journal of Mathematical Analysis and Applications, 435, No 1 (2016), 369-379.
  7. [7] B.A. Frasin, M. Darus, On the Fekete-Szeg¨o problem, International Journal of Mathematics and Mathematical Sciences, 24, No 9 (2000), 577-581.
  8. [8] S.A. Halim, R. Omar, Applications of certain functions associated with lemniscate Bernoulli, J. Indones. Math. Soc, 18, No 2 (2012), 93-99.
  9. [9] A. Janteng, S.A. Halim and M. Darus, Coefficient inequality for a function whose derivative has a positive real part, Journal of Inequalities in Pure and Applied Mathematics, 7, No 2 (2006), 1-5.
  10. [10] S. Kanas, An unified approach to the Fekete-Szeg¨o problem, Applied Mathematics and Computation, 218, No 17 (2012), 8453-8461.
  11. [11] S.K. Lee, V. Ravichandran and S. Supramaniam, Bounds for the second Hankel determinant of certain univalent functions, Journal of Inequalities and Applications, 2013, No 1 (2013), 281. ON A SUBCLASS OF STARLIKE FUNCTIONS... 781
  12. [12] R.J. Libera and E.J. Zlotkiewicz, Coefficient bounds for the inverse of a function with derivative in P, In: Proc. of the American Mathematical Society, 87, No 2 (1983), 251-257.
  13. [13] W. Ma, D. Minda, A unified treatment of some special classes of univalent functions, In: Proc. of the Conference on Complex Analysis, Int. Press (1994), 157-169.
  14. [14] T.H.Macgregor, Functions whose derivative has a positive real part, Transactions of the American Mathematical Society, 104, No 3 (1962), 532-537.
  15. [15] R. Mendiratta, S. Nagpal and V. Ravichandran, A subclass of starlike functions associated with left-half of the lemniscate of Bernoulli, International Journal of Mathematics 25, No 9 (2014), 1450090.
  16. [16] K.I. Noor, Hankel determinant problem for the class of functions with bounded boundary rotation, Revue Roumaine de Math´ematiques Pures et Appliqu´ees, 28, No 8 (1983), 731-739.
  17. [17] E. Paprocki, J. Sok´ol, The extremal problems in some subclass of strongly starlike function, Zeszyty Nauk. Politech. Rzeszowskiej Mat., 20 (1996), 89-94.
  18. [18] Ch. Pommerenke, On the coefficients and Hankel determinants of univalent functions, Journal of the London Mathematical Society 1 (1966), 111-122.
  19. [19] C. Ramachandran, D. Kavita, Toeplitz determinant for some subclasses of analytic functions, Global Journal of Pure and Applied Mathematics, 13, No 2 (2017), 785-793.
  20. [20] A.K. Sahoo, J.Patel, Hankel determinant for a class of analytic functions related with lemniscate of Bernoulli, International Journal of Analysis and Applications, 6, No 2 (2014), 170-177.
  21. [21] C. Selvaraj, T.R.K. Kumar, Second Hankel determinant for certain classes of analytic function, International Journal of Applied Mathematics, 28, No 1 (2015), 37-50; doi: 10.12732/ijam.v28i1.4.
  22. [22] K. Sharma, N.K. Jain and V. Ravichandran, Starlike functions associated with a cardioid, Afrika Matematika 27,(2016), 923-939.
  23. [23] K. Sharma, V. Ravichandran, Applications of subordination theory to starlike functions, Bull. Iranian Math. Soc., 42, No 3 (2016), 761-777. 782 Y. Yunus, A.B. Akbarally, S.A. Halim
  24. [24] S. Sivasubramanian, M. Govindaraj and G. Murugusundaramoorthy, Toeplitz matrices whose elements are the coefficients of analytic functions belonging to certain conic domains, International Journal of Pure and Applied Mathematics, 109, No 10 (2016), 39-49.
  25. [25] J. Sok´ol, Coefficient estimates in a class of strongly starlike functions, Kyungpook Mathematical Journal, 49, No 2 (2009).
  26. [26] J.Sok´ol, J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat., 19 (1996), 101-105.
  27. [27] D.K. Thomas, The second Hankel determinant for starlike functions of order alpha, arXiv Preprint, arXiv:1508.05839 (2015).
  28. [28] Y. Yunus, A.B. Akbarally and S.A. Halim, Properties of certain subclass of starlike functions defined by a generalized operator, International Journal of Applied Mathematics, 31, No 4 (2018), 597-611; doi: 10.12732/ijam.v3li46.
  29. [29] Y. Yunus, S.A. Halim and A.B. Akbarally,Subclass of starlike functions associated with a limacon, AIP Conference Proc. 1974 (2018), # 030023; doi: 10.1063/1.5041667.