Fekete Szegö properties for a class of univalent functions of complex order associated with q-difference operator


In this paper we introduce new class of q-starlike and q-convex functions of complex order involving the q-derivative operator. Morever, we find estimates on the coefficients for functions in this class.

This is a preview of subscription content, log in to check access.


  1. 1.

    Annby, M.H., Mansour, Z.S.: \(q\)-Fractional Calculas Equations. Lecture Noes in Math., vol. 2056. Springer, Berlin Heidelberg (2012)

  2. 2.

    Aouf, M.K., Darwish, H.E., Sălăgeăn, G.S.: On a generalaization of starlike functions with negative coefficients. Math. Tome 43(66), 3–10 (2001)

    MATH  Google Scholar 

  3. 3.

    Aouf, M.K., Seoudy, T.M.: Convolution properties for classes of bounded analytic functions with complex order defined by q-derivative operator. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. 113(2), 1279–1288 (2019)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Aral, A., Gupta, V., Agarwal, R.P.: Applications of q-Calculus in Operator Theory. Springer, New York (2013)

    Google Scholar 

  5. 5.

    Bulboacã, T.: Differential Subordinations and Superordinations, Recent Results. House of Science Book Publisher, Cluj-Napoca (2005)

    Google Scholar 

  6. 6.

    Deniz, E.: Certain subclasses of bi-univalent functions satisfying subordination condition. J. Class. Anal. 2(1), 49–60 (2013)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Frasin, B.A.: Family of analytic functions of complex order. Acta Math. Acad. Paedagog. Nyházi. (N. S.) 22(2), 179–191 (2006)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Gasper, G., Rahman, M.: Basic Hypergeometric Series. Combridge University Press, Cambridge (1990)

    Google Scholar 

  9. 9.

    Jackson, F.H.: On q-functions and a certain difference operator. Trans. R. Soc. Edinb. 46, 253–281 (1908)

    Article  Google Scholar 

  10. 10.

    Ma, W.C., Minda, D.: A unified treatment of some special classes of univalent functions. In: Proceedings of the Conference on Complex Analysis, Tianjin. Int. Press, Cambridge (1992)

  11. 11.

    Miller, S.S., Mocanu, P.T.: Differential Subordinations: Theory and Applications. Series on Monographs and Textbooks in Pure and Appl Math, vol. 225. Marcel Dekker Inc, New York (2000)

  12. 12.

    Nasr, M.A., Aouf, M.K.: On convex functions of complex order. Mansoura Bull. Sci. 8, 565–582 (1982)

    Google Scholar 

  13. 13.

    Nasr, M.A., Aouf, M.K.: Starlike functions of complex order. Nat. Sci. Math. 25, 1–12 (1985)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Ramachandran, C., Soupramanien, T., Frasin, B.: New subclasses of analytic functions associated with \(q-\)difference operator. Eur. J. Pure Appl. Math. 10(2), 348–362 (2017)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Ravichandran, V., Polatoglu, Y., Bolcal, M., Sen, A.: Certain subclasses of starlike and convex functions of complex order. Hacet. J. Math. Stat. 34, 9–15 (2005)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Robertson, M.S.: On the theory of univalent functions. Ann. Math. 37, 374–408 (1936)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Seoudy, T.M., Aouf, M.K.: Convolution properties for certain classes of analytic functions defined by q-derivative operator. Abstr. Appl. Anal. Article ID 846719 2014, 1–7 (2014)

  18. 18.

    Seoudy, T.M., Aouf, M.K.: Coefficient estimates of new classes of q-starlike and q-convex functions of complex order. J. Math. Inequal. 10(1), 135–145 (2016)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Srivastava, H.M., Mostafa, A.O., Aouf, M.K., Zayed, H.M.: Basic and fractional q-calculus and associated Fekete–Szegö problem for p-valently q-starlike functions and p-valently q-convex functions of complex order. Miskolc Math. Notes 20(1), 489–509 (2019)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Srivastava, H.M., Owa, S. (eds.): Current Topics in Analytic Function Theory. World Scientific Publishing Company, Singapore (1992)

    Google Scholar 

  21. 21.

    Tang, H., Zayed, H.M., Mostafa, A.O., Aouf, M.K.: Fekete-Szeg ö problems for certain classes of meromorphic functions using \(q-\) derivative operator. J. Math. Appl. 38(3), 236–246 (2018)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Wiatrowski, P.: On the coefficients of some family of holomorphic functions. Zesz. Nauk. Uniw. Lódz, Nauk. Mat.-Przyr. 39, 75–85 (1970)

    MATH  Google Scholar 

  23. 23.

    Zayed, H.M., Aouf, M.K.: Subclasses of analytic functions of complex order associated with q-Mittag Leffler function. J. Egyption Math. Soc. 26(2), 278–286 (2018)

    Article  Google Scholar 

Download references


The authors would like to thank the referees of the paper for their helpful suggestions.

Author information



Corresponding author

Correspondence to M. K. Aouf.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Aouf, M.K., Mostafa, A.O. & Elmorsy, R.E. Fekete Szegö properties for a class of univalent functions of complex order associated with q-difference operator. Afr. Mat. (2020). https://doi.org/10.1007/s13370-020-00807-z

Download citation


  • Analytic and univalent functions
  • q-starlike functions
  • q-convex functions
  • q-derivative operator
  • Subordination
  • Fekete–Szegö problem

Mathematics Subject Classification

  • 30C45