New Rates of Convergence for the Iterates of Some Positive Linear Operators

In this paper, we give quantitative results for the convergence of the iterates of some positive linear operators which preserve the functions e(0) and e(1), respectively, e(0) and e(2). We obtain estimates in terms of moduli of smoothness and we improve some previous results. We give examples for the q-Bernstein, the q-Bernstein-Durrmeyer, the q-Stancu, the q-Meyer-Konig-Zeller, the King type of the q-Bernstein and the King type of the q-genuine-Bernstein-Durrmeyer operators. We observe that some q-operators (0 < q < 1) provide better convergence for the iterates than the corresponding classical operators (q = 1).