The distance between two limit $q$-Bernstein operators

Abstract

For q ∈ (0, 1), let B q denote the limit q-Bernstein operator. In this paper, the distance between B q and B r for distinct q and r in the operator norm on C[0, 1] is estimated, and it is proved that 1 B q − B r 2, where both of the equalities can be attained. To elaborate more, the distance depends on whether or not r and q are rational powers of each other.For example, if r j = q m for all j, m ∈ N, then B q − B r = 2, and if r = q m , m ∈ N, then B q − B r = 2(m − 1)/m.