Approximation Properties of King Type \((p,\,q)\)-Bernstein Operators

  • Özge Dalmanoğlu
  • Mediha ÖrkcüEmail author
Research Paper


The present paper deals mainly with a King type modification of \(( p,\,q) \)-Bernstein operators. By improving the conditions given in Mursaleen et al. (On (p, q)-analogue of Bernstein operators. Appl Math Comput 266:874–882, 2015a), we investigate the Korovkin type approximation of both \(( p,\,q) \)-Bernstein and King type \(( p,\,q) \)-Bernstein operators. We also prove that the error estimation of King type of the operator is better than that of the classical one whenever \(0\le x\le \frac{1}{3}.\)


\((p,\,q)\)-integers \((p,\,q)\)-Bernstein Operators King type operators Rate of convergence 



We would like to thank the referees for their suggestions that improved the presentation of the paper.


  1. Acar T (2016) (p, q)-Generalization of Szász–Mirakyan operators. Math Methods Appl Sci 39(10):2685–2695MathSciNetCrossRefzbMATHGoogle Scholar
  2. Acar T, Aral A, Mohiuddine SA (2016) Approximation by bivariate (p, q)-Bernstein–Kantorovich operators. Iran J Sci Technol Trans A.
  3. Aral A, Gupta V, Agarwal RP (2013) Applications of q-calculus in operator theory. Springer, New YorkCrossRefzbMATHGoogle Scholar
  4. Cai QB (2017) On (p, q)-analogue of modified Bernstein–Schurer operators for functions of one and two variables. J Appl Math Comput 54(1–2):1–21MathSciNetCrossRefzbMATHGoogle Scholar
  5. Cai QB, Zhou G (2016) On (p, q)-analogue of Kantorovich type Bernstein–Stancu–Schurer operators. Appl Math Comput 276:12–20MathSciNetGoogle Scholar
  6. Gupta V (2016) (p, q)-Szász–Mirakyan–Baskakov operators. Complex Anal Oper Theory.
  7. Gupta V (2016) Bernstein Durrmeyer operators based on two parameters. Facta Univ Ser Math Inform 31(1):79–95MathSciNetzbMATHGoogle Scholar
  8. Gupta V (2016) (p, q)-Genuine Bernstein Durrmeyer operators. Boll Unione Mat Ital 9(3):399–409MathSciNetCrossRefzbMATHGoogle Scholar
  9. Kac VG, Cheung P (2002) Quantum calculus. Universitext. Springer, New YorkCrossRefzbMATHGoogle Scholar
  10. Karaisa A (2016) On the approximation properties of bivariate (p, q)-Bernstein operators. arXiv:1601.05250
  11. King JP (2003) Positive linear operators which preserve \( x^{2}\). Acta Math Hung 99(3):203–208CrossRefzbMATHGoogle Scholar
  12. Lupas A (1987) A \(q\)-analogue of the Bernstein operator. Univ Cluj-Napoca Semin Numer Stat Calc Prepr 9:85–92MathSciNetzbMATHGoogle Scholar
  13. Mahmudov N (2009) Korovkin-type theorems and applications. Open Math 7(2):348–356MathSciNetCrossRefzbMATHGoogle Scholar
  14. Mishra VN, Pandey S (2016) Chlodowsky variant of (p, q) Kantorovich–Stancu–Schurer operators. Int J Anal Appl 11(1):28–39MathSciNetzbMATHGoogle Scholar
  15. Mursaleen M, Ansari KJ, Khan A (2015) On (p, q)-analogue of Bernstein operators. Appl Math Comput 266:874–882MathSciNetzbMATHGoogle Scholar
  16. Mursaleen M, Nasiruzzaman M, Khan A, Ansari KJ (2016) Some approximation results on Bleimann–Butzer–Hahn operators defined by (p, q)-integers. Filomat 30(3):639–648MathSciNetCrossRefzbMATHGoogle Scholar
  17. Mursaleen M, Ansari KJ, Khan A (2015) On (p, q)-analogue of Bernstein operators (revised). arXiv preprint arXiv:1503.07404
  18. Mursaleen M, Ansari KJ, Khan A (2015) Some approximation results by (p, q)-analogue of Bernstein–Stancu operators. Appl Math Comput 264(2015):392–402 [Corrigendum Appl Math Comput 269:744–746]Google Scholar
  19. Mursaleen M, Nasiruzzaman M, Ashirbayev N (2015) Some approximation results on Bernstein–Schurer operators defined by (p, q)-integers. J Inequal Appl.
  20. Phillips GM (1996) Bernstein polynomials based on the q-integers. Ann Numer Math 4:511–518MathSciNetzbMATHGoogle Scholar

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© Shiraz University 2017

Authors and Affiliations

  1. 1.Department of Mathematics EducationBaskent UniversityAnkaraTurkey
  2. 2.Department of Mathematics, Faculty of SciencesGazi UniversityAnkaraTurkey

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