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The Youngnam Mathematical Society (영남수학회)
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BI-UNIVALENT FUNCTIONS OF COMPLEX ORDER BASED ON SUBORDINATE CONDITIONS INVOLVING HURWITZ-LERCH ZETA FUNCTION

  • Murugusundaramoorthy, G. (School of Advanced Sciences VIT University) ;
  • Janani, T. (School of Advanced Sciences VIT University) ;
  • Cho, Nak Eun (Department of Applied Mathematics Pukyong National University)
  • Received : 2015.11.19
  • Accepted : 2016.01.13
  • Published : 2016.01.30
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Abstract

The purpose of the present paper is to introduce and investigate two new subclasses of bi-univalent functions of complex order defined in the open unit disk, which are associated with Hurwitz-Lerch zeta function and satisfying subordinate conditions. Furthermore, we find estimates on the Taylor-Maclaurin coefficients ${\mid}a_2{\mid}$ and ${\mid}a_3{\mid}$ for functions in the new subclasses. Several (known or new) consequences of the results are also pointed out.

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References

  1. R. M. Ali, S.K. Lee, V. Ravichandran, S. Supramaniam, Coeffcient estimates for biunivalent Ma-Minda star-like and convex functions,Appl. Math. Lett. 25 (2012) 344-351. https://doi.org/10.1016/j.aml.2011.09.012
  2. J. W. Alexander, Functions which map the interior of the unit circle upon simple regions, Ann. Math. 17(1915), 12-22. https://doi.org/10.2307/2007212
  3. D. A. Brannan and T. S. Taha, On some classes of bi-univalent functions, Studia Univ. Babes-Bolyai Math. 31(2) (1986), 70-77.
  4. S. D. Bernardi, Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135 (1969), 429-446. https://doi.org/10.1090/S0002-9947-1969-0232920-2
  5. J. Choi and H. M. Srivastava, Certain families of series associated with the Hurwitz-Lerch Zeta function, Appl. Math. Comput. 170 (2005), 399-409.
  6. E. Deniz, Certain subclasses of bi-univalent functions satisfying subordinate conditions, Jour. Class. Anal. 2(1) (2013), 49-60.
  7. C. Ferreira and J. L. Lopez, Asymptotic expansions of the Hurwitz-Lerch Zeta function, J. Math. Anal. Appl., 298 (2004), 210-224. https://doi.org/10.1016/j.jmaa.2004.05.040
  8. T. M. Flett, The dual of an inequality of Hardy and Littlewood and some related inequalities, J. Math. Anal. Appl. 38 (1972), 746-765 https://doi.org/10.1016/0022-247X(72)90081-9
  9. B. A. Frasin and M. K. Aouf, New subclasses of bi-univalent functions, Appl. Math. Lett. 24 (2011), 1569-1573. https://doi.org/10.1016/j.aml.2011.03.048
  10. M. Garg, K. Jain and H. M. Srivastava, Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch Zeta functions, Integral Transform. Spec. Funct. 17 (2006), 803-815. https://doi.org/10.1080/10652460600926907
  11. T. Hayami and S. Owa, Coeffcient bounds for bi-univalent functions, Pan Amer. Math. J. 22(4) (2012), 15-26.
  12. I. B. Jung, Y. C. Kim AND H. M. Srivastava, The Hardy space of analytic functions associated with certain one-parameter families of integral operators, J. Math. Anal. Appl. 176 (1993), 138-147. https://doi.org/10.1006/jmaa.1993.1204
  13. R. J. Libera, Some classes of regular univalent functions, Proc. Amer. Math. Soc. 16 (1965), 755-758. https://doi.org/10.1090/S0002-9939-1965-0178131-2
  14. A. E. Livingston, On the radius of univalence of certain analytic functions, Proc. Amer. Math. Soc. 17 (1966), 352-357. https://doi.org/10.1090/S0002-9939-1966-0188423-X
  15. S.-D. Lin and H. M. Srivastava, Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput. 154 (2004), 725-733.
  16. S.-D. Lin, H. M. Srivastava and P.-Y. Wang, Some espansion formulas for a class of generalized Hurwitz-Lerch Zeta functions, Integral Transform. Spec. Funct. 17 (2006), 817-827. https://doi.org/10.1080/10652460600926923
  17. Y. Ling and F.-S. Liu, The Choi-Saigo-Srivastava integral operator and a class of analytic functions, Appl. Math. Comput. 165 (2005), 613-621.
  18. X.-F. Li and A.-P. Wang, Two new subclasses of bi-univalent functions, Internat. Math. Forum 7 (2012), 1495-1504.
  19. W.C. Ma, D. Minda, A unified treatment of some special classes of functions, in: Proceedings of the Conference on Complex Analysis, Tianjin, 1992, 157-169, Conf. Proc. Lecture Notes Anal. 1. Int. Press, Cambridge, MA, 1994.
  20. G.Murugusundaramoorthy, Subordination results for spirallike functions associated with Hurwitz-Lerch zeta function, Integral Transform. Spec. Funct. 23(2) (2012) 97-103 https://doi.org/10.1080/10652469.2011.562501
  21. G. Murugusundaramoorthy , N. Magesh and V.Praemala, Coeffcient bounds for certain subclasses of bi-univalent function, Abst. Appl. Anal. 2013, Article ID 573017, 3 pages.
  22. J. K. Prajapat and S. P. Goyal, Applications of Srivastava-Attiya operator to the classes of strongly starlike and strongly convex functions, J. Math. Inequal. 3 (2009), 129-?37.
  23. C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Gottingen, 1975.
  24. T. Panigarhi and G. Murugusundaramoorthy, Coeffcient bounds for Bi- univalent functions analytic functions associated with Hohlov operator, Proc. Jangjeon Math. Soc. 16 (1) (2013), 91-100.
  25. H. M. Srivastava and J. Choi, Series associated with the Zeta and related functions, Dordrecht, Boston, London: Kluwer Academic Publishers, 2001.
  26. H. M. Srivastava, A. K. Mishra and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Appl. Math. Lett. 23 (2010), 1188-1192. https://doi.org/10.1016/j.aml.2010.05.009
  27. H. M. Srivastava, G. Murugusundaramoorthy and N. Magesh,, Certain subclasses of biunivalent functions associated with the Hohlov operator, Global Jour. Math. Anal. 1(2) (2013) 67-73.
  28. H. M. Srivastava and A. A. Attiya, An integral operator associated with the Hurwitz-Lerch Zeta function and di erential subordination, Integral Transform. Spec. Funct. 18 (2007), 207-216. https://doi.org/10.1080/10652460701208577
  29. Q.-H. Xu, Y.-C. Gui and H. M. Srivastava, Coeffcient estimates for a certain subclass of analytic and bi-univalent functions, Appl. Math. Lett. 25 (2012), 990-994. https://doi.org/10.1016/j.aml.2011.11.013
  30. Q.-H. Xu, H.-G. Xiao and H. M. Srivastava, A certain general subclass of analytic and bi-univalent functions and associated coeffcient estimate problems, Appl. Math. Comput. 218 (2012), 11461-11465.