Approximation by (p,q) Szász-beta–Stancu operators

  • Prerna Maheshwari SharmaEmail author
  • Mohammed Abid
Open Access


Motivated by recent investigations, in this paper we introduce (pq)-Szász-beta–Stancu operators and investigate their local approximation properties in terms of modulus of continuity. We also obtain a weighted approximation and Voronovskaya-type asymptotic formula.

Mathematics Subject Classification

41A30 41A35 



  1. 1.
    Acar, T.: \((p, q)\) generalization of Szász-Mirakyan operators. Math. Method Appl. Sci 39(10), 2685–2695 (2016)CrossRefGoogle Scholar
  2. 2.
    Ansari, K.J.; Karaisa, A.: On the approximation by Chlodowsky type generalization of \((p, q)\)-Bernstien operators. Int. J. Nonlinear Anal. Appl. 8(2), 181–200 (2017)zbMATHGoogle Scholar
  3. 3.
    Aral, A.; Gupta, V.: \((p,q)\) variant of Szász Beta operators. Rev. de la Real Acad. de Cien. Exact. Fis Y Nat. Ser. A Mat, 111, 719–733 (2017)CrossRefGoogle Scholar
  4. 4.
    Aral, A.; Gupta, V.: \((p, q)\) type Beta functions of second kind. Adv. Oper. Theory 1(1), 134–146 (2016)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Aral, A.; Gupta, V.; Agarwal, R.P.: Applications of \(q\)-Calculus in Operator Theory. Springer, Berlin (2013)CrossRefGoogle Scholar
  6. 6.
    Devdhara, A.R.; Mishra, V.N.: Stancu variant of \((p, q)\) Szász Mirakyan operators. J. Ineq. Spec. Func. 8(5), 1–7 (2017)Google Scholar
  7. 7.
    Devore, R.A.; Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)CrossRefGoogle Scholar
  8. 8.
    Gadz̃hiev, A. D.: Theorem of the type of P. P Korovkin type theorems, presented at the international conference on the theory of approximation of functions (Kaluga, 1975). Mat. Zametki, 20(5), 781–786 (1976)Google Scholar
  9. 9.
    Gupta, V.; Aral, A.: Bernstein Durrmeyer operators based on two parameters. Facta Univ. Ser. Math. Inf. 31(1), 79–95 (2016)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kac, V.; Cheung, P.: Quantum Calculus, Universitext. Springer, New York (2002)CrossRefGoogle Scholar
  11. 11.
    Lupas, A.: A q-analogue of the Bernstein operators, seminar on numerical and statistical calculus. Univ. Cluj Napoca 9, 85–92 (1987)zbMATHGoogle Scholar
  12. 12.
    Maheshwari, P.: Approximation properties of certain \(q\)-genuine Szasz operators. Complex Anal. Oper. Theory 12(1), 27–36 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Maheshwari, P.; Sharma, D.: Approximation by q Baskakov–Beta–Stancu operators. Rend. Circ. Mat. Palermo 61, 297–305 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mahmudov, N.I.: On q-parametric Szász Mirakjan operators. Mediterr. J. Math. 7(3), 297–311 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Malik, N., Araci, S., Beniwal, M.: Approximation of Durrmeyer type operators depending on certain parameters. Abst. Appl. Anal. (2017), article ID5316150, 9Google Scholar
  16. 16.
    Mursleen, M.; Nasiruzzaman, M.; Ansari, K.J.; Alotaibi, A.: Generalized \((p, q)\)-Bleimann–Butzer–Hahn operators and some approximation results. J. Ineq. Appl. 2017, 310 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mursleen, M., Alotaibi, A., Ansari, K. J.: On a Kantrovich variant of \((p,q)\)-Szasz-Mirakjan operators. J. Func. Spaces, Volume 2016, Article ID 1035253, 9.
  18. 18.
    Mursleen, M., Ansari, K. J., Khan, A.: On \((p,q)\)-analogue of Bernstein operators, Appl. Math. Comput. 266, 874–882 (2015) [(Erratum: Appl. Math. Comput, 278 (2016), 70–71)] Google Scholar
  19. 19.
    Mursleen, M.; Ansari, K.J.; Khan, A.: On \((p, q)\)-analogue of Bernstein operators. Appl. Math. Comput. 266, 874–882 (2015)MathSciNetGoogle Scholar
  20. 20.
    Mursleen, M., Ansari, K.J., Khan, A.: Some approximation results by \((p,q)\)-analogue of Bernstein–Stancu operators. Appl. Math. Comput. 264, 392–402 (2015) (Corrigendum: Appl. Math. Comput. 269 (2015) 744–746) Google Scholar
  21. 21.
    Phillips, G.M.: Bernstein polynomials based on the \(q\)-integers. Ann. Numer. Math. 4, 511–518 (1997)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Prakash, O.; Sharma, D.; Maheshwari, P.: Certain generalized q-operators. Demon. Math. 3(48), 404–412 (2015)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Sadjang, P.N.: On the \((p,q)\)-Gamma and \((p,q)\)-Beta functions. arXiv:1506.07394v1 (2015)
  24. 24.
    Sadjang, P.N.: On fundamental theorem of \((p,q)\)-calculus and some \((p,q)\)-Taylor’s formulas. (2018)
  25. 25.
    Sahai, V.; Yadav, S.: Repersentation of two parametric quantum algebra and \((p, q)\)-special functions. J. Math. Anal. Appl. 335, 268–279 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Basic SciencesSardar Vallabh Bhai Patel University of Agriculture and TechnologyMeerutIndia
  2. 2.Department of MathematicsSRM Institute of Science and Technology Delhi-NCR CampusModinagarIndia

Personalised recommendations