The distance between two limit $q$-Bernstein operators

Abstract: For $q\in(0,1),$ 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 $1\leqslant |B_q-B_r|\leqslant 2,$ 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 $r^j\neq q^m$ for all $j,m\in \mathbb{N},$ then $|B_q-B_r|=2,$ and if $r=q^m, m\in \mathbb{N},$ then $|B_q-B_r|=2(m-1)/m.$