The distance between two limit $q$-Bernstein operators

Abstract

For $q\in \left(0,1\right)$, let $B_q$ denote the limit $q$-Bernstein operator. The distance between $B_q$ and $B_r$ for distinct $q$ and $r$ in the operator norm on $C\left[0,1\right]$ is estimated, and it is proved that $1\le \parallel B_q-B_r\parallel \le 2$, where both of the equalities can be attained. Furthermore, the distance depends on whether or not $r$ and $q$ are rational powers of each other. For example, if $r^j\ne q^m$ for all $j,m\in \mathbb{N}$, then $\parallel B_q-B_r\parallel =2$, and if $r=q^m$ for some $m\in \mathbb{N}$, then $\parallel B_q-B_r\parallel =2\left(m-1\right)∕m$.