Convergence in Variation for Bernstein-Type Operators

In this paper, we deal with Bernstein-type operators defined by Cárdenas-Morales et al. as \({B_{n}(f \circ \tau^{-1}) \circ \tau}\), where \({B_{n}}\) is the nth Bernstein polynomial (Comput Math Appl 62(1):158–163, 2011). Assuming that \({\tau}\) and f are absolutely continuous functions on \({0, 1]}\) and inf \({\tau ^{\prime} (x) \geq m > 0}\) as well as \({\tau (0) = 0}\) and \({\tau (1) = 1,}\) we study the convergence of Bernstein-type operators to f in variation seminorm. Moreover, we give a Voronovskaja-type formula and a Jackson-type estimate in the sense of Bardaro et al. (Analysis 23:299–340, 2003).