On Kantorovich Modification of (pq)-Bernstein Operators

  • Tuncer Acar
  • Ali Aral
  • S. A. MohiuddineEmail author
Research Paper


In the present paper, we introduce Kantorovich modifications of (pq)-Bernstein operators using a new (pq) -integral. We first estimate the moments and central moments. We obtain uniform convergence of new operators, rate of convergence in terms of classical modulus of continuity and second order modulus of continuity. We also investigate the rate of convergence of new operators for functions belonging to Lipschitz class and finally, we give an upper bound for the error of approximation via modulus of continuity of the derivative of approximating function.


\((p, q)\)-Integers \((p, q)\)-Integral \((p, q)\)-Bernstein operators \((p, q)\)-Bernstein–Kantorovich operators Rate of convergence 



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


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© Shiraz University 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science and ArtsKirikkale UniversityKirikkaleTurkey
  2. 2.Operator Theory and Applications Research Group, Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia

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