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Mediterranean Journal of Mathematics

, Volume 13, Issue 4, pp 1535–1553 | Cite as

On the Convolution of a Finite Number of Analytic Functions Involving a Generalized Srivastava–Attiya Operator

  • Poonam Sharma
  • Ravinder Krishna Raina
  • Janusz Sokół
Article

Abstract

The present paper gives several subordination results involving a generalized Srivastava–Attiya operator (defined below). Among the results presented in this paper include also a sufficiency condition for the convexity of the convolution of certain functions and a sharp result relating to the convolution structure. We also mention various useful special cases of the main results including those which are related to the Zeta function.

Mathematics Subject Classification

Primary 30C45 30C50 

Keywords

Analytic functions Convolution subordination convex functions Zeta function 

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References

  1. 1.
    Abramowitz, M., Stegun, (Editors), I.A.: Handbook of Mathematical Functions and Formulas, Graphs and Mathematical Tables. Dover publications, New York (1971)Google Scholar
  2. 2.
    Cho, N.E., Srivastava, H.M.: Argumet estimate of certain analytic functions defined by a class of multiplier transformations, Math. Comput. Model. 37(2), 39–49 (2003)Google Scholar
  3. 3.
    Choi J.H., Saigo M., Srivastava H.M.: Some inclusion properties of a certain family of integral operators. J. Math. Anal. Appl. 276, 432–445 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Hallenbeck D.J., Ruscheweyh St.: Subordination by convex functions. Proc. Am. Math. Soc. 52, 191–195 (1975)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Janowski Z.: Some extremal problems for certain families of analytic functions. Ann. Polon. Math. 28, 297–326 (1973)MathSciNetMATHGoogle Scholar
  6. 6.
    Jung I.B., Kim Y.C., Srivastava H.M.: The Hardy space of analytic functions associated with certain one-parameter families of integral operators. J. Math. Anal. Appl. 176, 138–147 (1993)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Kwon O.S., Cho N.E.: Inclusion properties for a certain subclasses of analytic functions defined by a multiplier transformation. Eur. J. Pure Appl. Math. 3(6), 1124–1136 (2010)MathSciNetMATHGoogle Scholar
  8. 8.
    Miller, S.S., Mocanu, P.T.: Differential subordinations: theory and applications, Series of Monographs and Text books in Pure and Applied Mathematics, vol.~225. Marcel Dekker Inc., New York (2000)Google Scholar
  9. 9.
    Noor K.I., Bukhari S.Z.H.: Some subclasses of analytic and spiral-like functions of complex order involving the Srivastava–Attiya integral operator. Integral Transforms Spec. Funct. 21(12), 907–916 (2010)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Prajapat J.K., Bulboacă T.: Double subordination preserving properties for a new generalized Srivastava–Attiya operator. Chin. Ann. Math. 33, 569–582 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Prajapat J.K., Raina R.K.: Some applications of differential subordinatio to a general class of multivalently analytic functions involving a convolution structure. Bull. Math. Anal. Appl. 1(1), 1–14 (2009)MathSciNetMATHGoogle Scholar
  12. 12.
    Prajapat J.K., Raina R.K., Sokół J.: On a Hurwitz–Lerch Zeta type function and its applications. Bull. Belg. Math. Soc. Simon Stevin. 20, 803–820 (2013)MathSciNetMATHGoogle Scholar
  13. 13.
    Raina R.K., Sharma P.: Subordination properties of univalent functions involving a new class of operators. Electron. J. Math. Anal. Appl. 2(1), 37–52 (2014)MathSciNetGoogle Scholar
  14. 14.
    Ruscheweyh St., Stankiewicz J.: Subordination under convex univalent function. Bull. Pol. Acad. Sci. Math. 33, 499–502 (1985)MathSciNetMATHGoogle Scholar
  15. 15.
    Sharma P., Maurya R.K.: Certain subordination results on the convolution of analytic functions. J. Math. Appl. 37, 107–114 (2014)MathSciNetGoogle Scholar
  16. 16.
    Sharma P., Prajapat J.K., Raina R.K.: Certain subordination results involving a generalized multiplier transformation operator. J. Classical Anal. 2(1), 85–106 (2013)MathSciNetGoogle Scholar
  17. 17.
    Sharma P., Raina R.K., Sok ół J.: On the convolution of analytic functions involving a multiplier operator. Jokull J. 65(1), 231–253 (2015)Google Scholar
  18. 18.
    Sokół J.: The convexity of Hadamard product of three functions. J. Math. Appl. 29, 121–125 (2007)MathSciNetMATHGoogle Scholar
  19. 19.
    Srivastava, H.M., Owa S. (eds.): Current Topics in Anaytic Function Theory, pp. 266–273. World Scientific Publishing Company, Singapore (1992)Google Scholar
  20. 20.
    Srivastava, H.M., Karlsson, P.W.: Multiple Gaussian Hypergeometric Series. Halsted Press/Wiley, Chichester/New York (1985)Google Scholar
  21. 21.
    Srivastava H.M., Attiya A.A.: An integral operator associated with the Hurwitz–Lerch Zeta function and differential subordination. Integral Transforms Spec. Funct. 18, 207–216 (2007)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Srivastava, H.M., Choi, J.: Series Associated with the Zeta and Related Functions. Kluwer Academic Publishers, Dordrecht (2001)Google Scholar
  23. 23.
    Srivastava H.M.: A new family of the k-generalized Hurwitz–Lerch Zeta functions with applications. Appl. Math. Inform. Sci. 8, 1485–1500 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Srivastava H.M., Gaboury S., Ghanim F.: A unified class of analytic functions involving a generalization of the Srivastava–Attiya operator. Appl. Math. Comput. 251, 35–45 (2015)MathSciNetMATHGoogle Scholar
  25. 25.
    Srivastava H.M., Saxena R.K., Pogány T.K., Saxena R.: Integral and computational representations of the extended Hurwitz–Lerch Zeta function. Integral Transforms Spec. Funct. 22, 487–506 (2011)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Stankiewicz J., Stankiewicz Z.: Some applications of the Hadamard convolution in the theory of functions. Ann. Univ. Mariae Curie-Skłodowska 40, 251–265 (1986)MathSciNetMATHGoogle Scholar
  27. 27.
    Stein E.M., Shakarchi R.: Complex Analysis. Princeton University Press, Princeton (2003)MATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Poonam Sharma
    • 1
  • Ravinder Krishna Raina
    • 2
    • 3
  • Janusz Sokół
    • 4
  1. 1.Department of Mathematics and AstronomyUniversity of LucknowLucknowIndia
  2. 2.M.P. University of Agriculture and TechnologyUdaipurIndia
  3. 3.UdaipurIndia
  4. 4.Department of MathematicsRzeszów University of TechnologyRzeszówPoland

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