Approximation by Bivariate (pq)-Bernstein–Kantorovich Operators

  • Tuncer Acar
  • Ali Aral
  • S. A. Mohiuddine
Research Paper


In the present paper, we introduce Kantorovich modifications of (pq)-Bernstein operators for bivariate functions using a new (pq)-integral. We first estimate the moments and central moments. We give the uniform convergence of new operators, rate of convergence in terms of modulus of continuity. The approximations behaviours of the operators for functions having continuous partial derivatives and for functions belong to Lipschitz class are investigated as well.


\((p , q)\)-integers Bivariate \((p , q)\)-integral Bivariate \((p , q)\)-Bernstein–Kantorovich operators Rate of convergence Uniform convergence 



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


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Copyright information

© Shiraz University 2016

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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