Multivalently meromorphic functions with two fixed points defined by Srivastava-Attiya operator

  • Authors

    • Kaliyappan Vijaya Professor of Mathematics, School of Advanced Sciences, VIT University, Vellore-14.
    • Murugesan Kasthuri School of Advanced Sciences, VIT University, Vellore-14.
    2014-08-19
    https://doi.org/10.14419/gjma.v2i3.3176
  • Abstract

    In the present investigation we definea new class of meromorphic functions on the punctured unit disk $\mathcal{U}^*:= \{z\in \mathbb{C}: 0<|z|<1\}$ by making use of theSrivastava-Attiya operator. Coefficient inequalities, growth and distortion inequalities, aswell as  radii of meromorphically  starlikeness are obtained. We also  establish someresults concerning the convolution products and inclusion results.

    Keywords: Convolution(Hadamard product),Hurwitz-Learch zeta function, Integral operator, Meromorphic functions,Starlike function,Convex function.

     

  • References

    1. J. W. Alexander, "Functions which map the interior of the unit circle upon simple regions", Ann. Math. Ser.2, 17 (1915), pp.12-22.
    2. M. K. Aouf, "On a certain class of meromorphic univalent functions with positive coefficients", Rend. Mat. Appl. 7 (11), (1991), no. 2, pp. 209-219.
    3. M. K. Aouf and H. M. Hossen, "New criteria for meromorphic p-valent starlike functions", Tsukuba J. Math. 17(1993), pp. 481-486.
    4. M. K. Aouf, H. M. Hossen and H. E. Elattar, "A certain class of meromorphic multivalent functions with positive and fixed second coefficients", Punjab Univ. J. Math. 33(2000), pp.115-124.
    5. S. K. Bajpai, " A note on a class of meromorphic univalent functions", Rev. Roumaine Math. Pures Appl. 22 (1977), no. 3, pp.295-297.
    6. S. D. Bernardi, "Convex and starlike univalent functions", Trans. Amer. Math. Soc., 135 (1969), pp.429-446.
    7. N. E. Cho, S. H. Lee and S. Owa, " A class of meromorphic univalent functions with positive coefficients", Kobe J.Math., 4 (1987), no. 1, pp.43-50.
    8. J. Choi and H. M. Srivastava, " Certain families of series associated with the Hurwitz-Lerch Zeta function", Appl. Math. Comput., 170 (2005), pp.399-409.
    9. J. Clunie, " On meromorphic schlicht functions", J. London Math. Soc., 34 (1959), pp.215-216.
    10. C. Ferreira and J. L. Lopez, "Asymptotic expansions of the Hurwitz-Lerch Zeta function", J. Math. Anal. Appl.,298(2004), pp.210-224.
    11. T. M. Flett, "The dual of an inequality of Hardy and Littlewood and some related inequalities", J. Math. Anal. Appl., 38(1972), pp.746-765.
    12. 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), pp.803-815.
    13. 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), pp.725-733.
    14. S.-D. Lin, H. M. Srivastava and P.-Y. Wang, "Some expansion formulas for a class of generalized Hurwitz-Lerch Zeta functions", Integral Transform. Spec. Funct., 17(2006), pp.817-827.
    15. Joshi, S. B. and M. K. Aouf, " Meromorphic multivalent functions with positive and fixed second coefficients", Kyungpook Math. J., 35 (1995), pp.163-169.
    16. Joshi, S. B. and H. M. Srivastava, " A certain family of meromorphically multivalent functions", Comput. Math. Appl.38, no. 3-4,(1999) pp.201-211.
    17. 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), pp.138-147.
    18. Kulkarni, S. R., U. H. Naik and H. M. Srivastava, "A certain class of meromorphically p-valent quasi-convex functions", Pan Amer. Math. J., 8, no. 1,(1998) pp.57-64.
    19. R. J. Libera, "Some classes of regular univalent functions", Proc. Amer. Math. Soc., 16 (1969), pp.755-758.
    20. J-L. Liu, "Some Inclusion Properties for Certain Subclass of Meromorphically Multivalent Functions Involving the Srivastava-Attiya Operator", Tamsui Oxford Journal of Information and Mathematical Sciences. 28(3) (2012) 267-279.
    21. J.-L. Liu, H.M. Srivastava, "Classes of meromorphically multivalent functions associated with the generalized hypergeometric function", Math. Comput. Modelling 39 (2004) pp.21-34.
    22. J.-L. Liu, H.M. Srivastava, "Subclasses of meromorphically multivalent functions associated with a certain linear operator", Math. Comput. Modelling 39 (2004) pp.35-44.
    23. M. L. Mogra, T. R. Reddy, O. P. Juneja, "Meromorphic univalent functions with positive coefficients", Bull. Austral. Math. Soc., 32 (1985), pp.161-176.
    24. M. Darus, "Meromorphic functions with positive coefficients", Int. J. Math. Math. Sci., 4 (2004), pp.319-324.
    25. Montel. P., "Lecons sur les Fonctions Univalentes ou Multivalentes", Gauthier-Villars, Paris (1933).
    26. Z. Nehari, E. Netanyahu, " On the coefficients of meromorphic schlicht functions", Proc. Amer. Math. Soc., 8 (1957), pp.15-23.
    27. Owa, S., and N. N. Pascu, " Coefficient inequalities for certain classes of meromorphically starlike and meromorphically convex functions", J. Inequal. Pure Appl. Math., 4, no. 1, (2003) Article 17, pp.1-6.
    28. Ch. Pommerenke, " ber einige Klassen meromorpher schlichter Funktionen", Math. Z.(German), 78 (1962), pp.263-284.
    29. Ch. Pommerenke, " On meromorphic starlike functions", Pacific J. Math., 13 (1963), pp.221-235.
    30. Raina, R. K. and H. M. Srivastava, " A new class of meromorphically multivalent functions with applications to generalized hypergeometric functions", Math. Comput. Modelling, 43,(2006) pp.350-356.
    31. W. C. Royster, " Meromorphic starlike multivalent functions", Trans. Amer. Math. Soc., 107 (1963), pp.300-308.
    32. A. Schild and H. Silverman, "Convolution of univalent functions with negative coefficients", Ann. Univ. Mariae-Curie Sklodowska, Sect. .A 29 (1975), pp.99-107.
    33. H. M. Srivastava, S. Owa (eds.), " Current Topics in Analytic Function Theory", World Scientific Publishing, New Jersey, 1992.
    34. H. M. Srivastava and A. A. Attiya, "An integral operator associated with the Hurwitz-Lerch Zeta function and differential subordination", Integral Transform. Spec. Funct., 18(2007), pp.207-216.
    35. H. M. Srivastava , H. M. Hossen and M. K. Aouf, "A unified presentation of some classes of meromorphically multivalent functions", Comput. Math. Appl., 38(1999), pp.63-70.
    36. B. A. Uralegaddi, M. D. Ganigi, "A certain class of meromorphically starlike functions with positive coefficients", Pure Appl. Math. Sci., 26(1987), pp.75-81.
    37. B. A. Uralegaddi and C. Somanatha, " Certain differential operators for meromorphic functions", Houston J. Math., 17 (1991), no. 2, pp.279-284.
    38. K.Vijaya,G.Murugusundaramoorthy and P.Kathiravan " Multivalently meromorphic functions associated with convolution structure", Acta Universitatis Apulensis, No. 34(2013) pp. 247-263
    39. Yang, D.G., "Subclasses of meromorphic p-valent convex functions", J. Math. Res. Exposition 20(2000), pp.215-219.
  • Downloads

  • How to Cite

    Vijaya, K., & Kasthuri, M. (2014). Multivalently meromorphic functions with two fixed points defined by Srivastava-Attiya operator. Global Journal of Mathematical Analysis, 2(3), 213-226. https://doi.org/10.14419/gjma.v2i3.3176

    Received date: 2014-07-14

    Accepted date: 2014-08-10

    Published date: 2014-08-19