On the rates of convergence of BBH-Kantorovich operators and their Bézier variant

Abstract

In the present paper we consider the Bézier variant of BBH-Kantorovich operators Jn,αf for functions f measurable and locally bounded on the interval [0,∞) with α⩾1. By using the Chanturiya modulus of variation we estimate the rate of pointwise convergence of Jn,αf(x) at those x>0 at which the one-sided limits f(x+), f(x−) exist. The very recent result of Chen and Zeng (2009) [L. Chen, X.M. Zeng, Rate of convergence of a new type Kantorovich variant of Bleimann–Butzer–Hahn Operators, J. Inequal. Appl. 2009 (2009) 10. Article ID 852897] is extended to more general classes of functions.