Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Fekete-Szegö Problem and Upper Bound of Second Hankel Determinant for a New Class of Analytic Functions

  • Bansal, Deepak (Department of Mathematics, Government College of Engineering and Technology)
  • Received : 2012.08.04
  • Accepted : 2014.02.24
  • Published : 2014.09.23
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Abstract

In the present investigation we consider Fekete-Szeg$\ddot{o}$ problem with complex parameter ${\mu}$ and also find upper bound of the second Hankel determinant ${\mid}a_2a_4-a^2_3{\mid}$ for functions belonging to a new class $S^{\tau}_{\gamma}(A,B)$ using Toeplitz determinants.

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References

  1. D. Bansal, Fekete-Szego problem for a new class of analytic functions, Int. J. Math. Math. Sci., (2011), doi:10.1155/2011/143096.
  2. D. Bansal, Upper bound of second Hankel determinant for a new class of analytic functions, Appl. Math. Lett., (2012), doi:10.1016/j.aml.2012.04.002.
  3. K. K. Dixit and S. K. Pal, On a class of univalent functions related to complex order, Indian J. Pure Appl. Math., 26(9)(1995), 889-896.
  4. P. L. Duren, Univalent Functions, vol. 259 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983.
  5. M. Fekete and G. Szego, Eine bemerkung uber ungerade schlichten funktionene, J. London. Math. Soc., 8(1993), 85-89.
  6. A. Gangadharan, T. N. Shanmugam and H. M. Srivastava, Generalized hypergeometric functions associate with k-uniformly convex functions, Comput. Math. Appl., 44(2002), 1515-1526. https://doi.org/10.1016/S0898-1221(02)00275-4
  7. U. Grenanderand G. Szego, Toeplitz forms and their application, Univ. of California Press, Berkeley and Los Angeles, 1958.
  8. A. Janteng, S. A. Halim and M. Darus, Hankel determinant for starlike and convex functions, Int. J. Math. Anal., 1(13)(2007), 619-625.
  9. A. Janteng, S. A. Halim and M. Darus, Coefficient inequality for a function whose derivative has a positive real part, J. Ineq. Pure Appl. Math., 7(2)(2006), Art. 50, 1-5.
  10. F. R. Keogh and E. P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc., 20(1969), 8-12. https://doi.org/10.1090/S0002-9939-1969-0232926-9
  11. Y. C. Kim and F. Rnning, Integral tranform of certain subclasses of analytic functions, J. Math. Anal. Appl., 258(2001), 466-486. https://doi.org/10.1006/jmaa.2000.7383
  12. W. Koepf, On the Fekete-Szego problem for close-to-convex functions, Proc. Amer. Math. Soc., 101(1)(1987), 89-95.
  13. W. Koepf, On the Fekete-Szego problem for close-to-convex functions II, Archiv der-Mathematik, 49(5)(1987), 420-433. https://doi.org/10.1007/BF01194100
  14. R. J. Libera and E. J. Zlotkiewicz, Coefficient bounds for the inverse of a function with derivative in P, Proc. Amer. Math. Soc., 87(2)(1983), 251-257.
  15. R. R. London, Fekete-Szego inequalities for close-to-convex functions, Proc. Amer. Math. Soc., 117(4)(1993), 947-950.
  16. W. C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, In Proceedings of the Conference on Complex Analysis, Z. Li, F. Ren, L. Lang, and S. Zhang (Eds.), Int. Press (1994), 157-169.
  17. A. K. Mishra and P. Gochhayat, Second Hankel Determinant for a class of Analytic Functions Defined by Fractional Derivative, Int. J. Math. Math. Sci., (2008), doi:10.1155/2008/153280.
  18. A. K. Mishra and S. N. Kund, Second Hankel determinant for a class of functions defined by the Carlson-Shaffer operator, Tamkang J. Math., 44(1)(2013), 73-82.
  19. J. W. Noonan and D. K. Thomas, On the second Hankel determinant of areally mean p-valent functions, Trans. Amer. Math. Soc., 223(2)(1976), 337-346.
  20. S. Ponnusamy and F. Rnning, Integral transform of a class of analytic functions, Complex Var. Elliptic Equ., 53(5)(2008), 423-434. https://doi.org/10.1080/17476930701685767
  21. S. Ponnusamy, Neighborhoods and Caratheodory functions, J. Anal., 4(1996), 41-51.
  22. R. K. Raina and D. Bansal, Integral means inequalities, neighborhood properties and other properties involving certain integral operators for a class of rational functions, J. Inequal. Pure Appl. Math., 9(2008), 1-9.
  23. W. Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc., 48(1943), 48-82.
  24. H. M. Srivastava, A. K. Mishra and M. K. Das, Fekete-Szego problem for a subclass of close-to-convex functions, Complex Var. Theory Appl., 44(2001), 145-163. https://doi.org/10.1080/17476930108815351
  25. A. Swaminathan, Certain sufficiency conditions on Gaussian hypergeometric functions, J. Ineq. Pure Appl. Math., 5(4)(2004), Art.3, 1-10.

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