Complex Analysis and Operator Theory

, Volume 12, Issue 6, pp 1453–1468 | Cite as

Approximation by (pq)-Baskakov–Durrmeyer–Stancu Operators

  • Tuncer Acar
  • S. A. Mohiuddine
  • Mohammad Mursaleen


The present paper deals with the Stancu-type generalization of (pq)-Baskakov–Durrmeyer operators. We investigate local approximation, weighted approximation properties of new operators and present the rate of convergence by means of suitable modulus of continuity. At the end of the paper, we introduce a new modification of (pq)-Baskakov–Durrmeyer–Stancu operators with King approach.


\((p , q)\)-Integers \((p , q)\)-Baskakov–Durrmeyer operators Weighted approximation 

Mathematics Subject Classification

41A25 41A35 41A36 



The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.


  1. 1.
    Acar, T.: \((p, q)\)-generalization of Szász–Mirakyan operators. Math. Methods Appl. Sci. 39(10), 2685–2695 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Acar, T., Agrawal, P.N., Kumar. S.: On a modification of (p,q)-Szász-Mirakyan operators. Complex Anal. Oper. Theory. (2016). doi: 10.1007/s11785-016-0613-9
  3. 3.
    Acar, T., Aral, A., Mohiuddine, S.A.: On Kantorovich modifications of \((p, q)\)-Baskakov operators. J. Inequal. Appl. 98, 2016 (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Acar, T., Aral, A., Mohiuddine, S.A.: Approximation By Bivariate (p,q)-Bernstein-Kantorovich operators, Iran. J. Sci. Technol. Trans. Sci. (2016). doi: 10.1007/s40995-016-0045-4
  5. 5.
    Aral, A., Gupta, V.: \((p, q)\)-type beta functions of second kind. Adv. Oper. Theor. 1(1), 134–146 (2016)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Aral, A., Gupta, V., Agarwal, R.P.: Applications of q-Calculus in Operator Theory. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  7. 7.
    Burban, I.: Two-parameter deformation of the oscillator albegra and \((p, q) \) analog of two dimensional conformal field theory. Nonlinear Math. Phys. 2(3–4), 384–391 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Burban, I.M., Klimyk, A.U.: \(P, Q\) differentiation, \(P, Q\) integration and \(P, Q\) hypergeometric functions related to quantum groups. Integr. Transforms Spec. Fucnt. 2(1), 15–36 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Devore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)CrossRefzbMATHGoogle Scholar
  10. 10.
    Ibikli, E., Gadjieva, E.A.: The order of approximation of some unbounded function by the sequences of positive linear operators. Turk. J. Math. 19(3), 331–337 (1995)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Ilarslan, H.G.I., Acar, T.: Approximation by bivariate (p,q)-Baskakov–Kantorovich operators, Georgian Mathemath. J. (2016). doi: 10.1515/gmj-2016-0057
  12. 12.
    King, J.P.: Positive linear operators which preserves \(x^{2}\). Acta Math. Hung. 99, 203–208 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lopez-Moreno, A.J.: Weighted simultaneous approximation with Baskakov type operators. Acta Math. Hung. 104(1–2), 143–151 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lupas, A.: A \(q\)-analogue of the Bernstein operator. Semin. Numer. Stat. Calc. Univ. Cluj Napoca 9, 85–92 (1987)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Mursaleen, M., Ansari, K.J., Khan, A.: On \((p, q)\)-analogue of Bernstein operators. Appl. Math. Comput. 266, 874–882 (2015)MathSciNetGoogle Scholar
  16. 16.
    Mursaleen, 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, 744–746 (2015)]Google Scholar
  17. 17.
    Mursaleen, M., Nasiruzzaman, Md, Khan, A., Ansari, K.J.: Some approximation results on Bleimann–Butzer–Hahn operators defined by \((p, q)\)-integers. Filomat 30, 639–648 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mursaleen, M., Nasiuzzaman, Md, Nurgali, A.: Some approximation results on Bernstein–Schurer operators defined by \((p, q)\)-integers. J. Inequal. Appl. 2015, 249 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sahai, V., Yadav, S.: Representations of two parameter quantum algebras and \(p, q\)-special functions. J. Math. Anal. Appl. 335, 268–279 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  • Tuncer Acar
    • 1
  • S. A. Mohiuddine
    • 2
  • Mohammad Mursaleen
    • 3
  1. 1.Department of Mathematics, Faculty of Science and ArtsKirikkale UniversityYahsihanTurkey
  2. 2.Operator Theory and Applications Research Group, Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia
  3. 3.Department of MathematicsAligarh Muslim UniversityAligarhIndia

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