On \(\alpha \)-Convex Multivalent Functions Defined by Generalized Ruscheweyh Derivatives Involving Fractional Differential Operator

  • Ritu Agarwal
  • J. Sokol
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 236)


In the present investigation, we introduce a class of \(\alpha \)-convex multivalent functions defined by generalized Ruscheweyh derivatives introduced by Goyal and Goyal (J. Indian Acad. Math. 27(2):439–456, 2005) which involves a generalized fractional differential operator. The necessary and sufficient condition for functions to belong to this class is obtained. We study properties of this class and derive a theorem about image of a function from this class through generalized Komatu integral operator. Also, the integral representation for the functions of this class has been obtained.


  1. 1.
    Acu, M., et al.: On some \(\alpha \)-convex functions. Int. J. Open Problems Comput. Sci. Math 1(1), 1–10 (2008).Google Scholar
  2. 2.
    Ali, R.M., Ravichandran, V.: Classes of meromorphic \(\alpha \)-convex functions. Taiwanese J. Math. 14(4), 1479–1490 (2010)MATHMathSciNetGoogle Scholar
  3. 3.
    Duren, P.L.: Univalent funct. Springer Verlag, New York (1983)Google Scholar
  4. 4.
    Eenigenburg, P.J., Miller, S.S., Mocanu, P.T., Reade, M.O.: On a Briot-Bouquet differential subordination. General inequalities, 3. Int. Series of Numerical Math, pp. 339–348. Birkhauser Verlag, Basel (1983).Google Scholar
  5. 5.
    Goyal, S.P., Goyal, R.: On a class of multivalent functions defined by generalized Ruscheweyh derivatives involving a general fractional derivative operator. J. Indian Acad. Math. 27(2), 439–456 (2005)Google Scholar
  6. 6.
    Komatu, Y.: On analytic prolongation of a family of operators. Mathematica (Cluj) 32(55), 141–145 (1990)MathSciNetGoogle Scholar
  7. 7.
    Ma, W., Minda, D.: A unified treatment of some special classes of univalent functions, proceedings of the conference on complex analysis, Tianjin, China. Int. Press. Conf. Proc. Lect. Notes Anal. 1, 157–169 (1994).Google Scholar
  8. 8.
    Miller, S.S., Mocanu, P.T.: On some classes of first-order differential subordinations. Michig. Math. J. 32, 185–195 (1985)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Mocanu, P.T.: Une propriete de convexite geneealisee dans la theorie de la representation conforme. Mathematica (Cluj) 11(34), 127–133 (1969)MathSciNetGoogle Scholar
  10. 10.
    Parihar, H.S., Agarwal, R.: Application of generalized Ruscheweyh derivatives on p-valent functions. J. Math. Appl. 34, 75–86 (2011)MathSciNetGoogle Scholar
  11. 11.
    Raducanu, D., Nechita, V.O.: On \(\alpha \)-convex functions defined by generalized Ruscheweyh derivatives operator, Studia Univ. "Babes-Bolyai". Mathematica 53, 109–118 (2008)MATHMathSciNetGoogle Scholar
  12. 12.
    Ravichandran, V., Darus, M.: On class of \(\alpha \)-convex functions. J. Anal. Appl. 2(1), 17–25 (2004)MATHMathSciNetGoogle Scholar
  13. 13.
    Ravichandran, V., Seenivasagan, N., Srivastava, H.M.: Some inequalities associated with a linear operator defined for a class of multivalent functions. J. Inequal. Pure and Appl. Math. 4(4), 1–7 (2003). Art. 70.Google Scholar
  14. 14.
    Ruscheweyh, S.T.: New criterion for univalent functions. Proc. Amer. Math. Soc. 49, 109–115 (1975)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Sokol, J.: On a condition for alpha-starlikeness. J. Math. Anal. Appl. 352, 696–701 (2009)Google Scholar
  16. 16.
    Srivastava, H.M., Saxena, R.K.: Operators of fractional integration and their applications. Appl. Math. Comput. 118, 1–52 (2001)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer India 2014

Authors and Affiliations

  1. 1.Malaviya National Institute of TechnologyJaipurIndia
  2. 2.Institute of MathematicsUniversity of RzeszówRzeszówPoland

Personalised recommendations