The distance between two limit q-Bernstein operators

For q ∈ (0, 1), let Bq denote the limit q-Bernstein operator. In this paper, the distance between Bq and Br for distinct q and r in the operator norm on C[0, 1] is estimated, and it is proved that 1 6 ‖Bq −Br‖ 6 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 rj 6= qm for all j,m ∈ N, then ‖Bq − Br‖ = 2, and if r = qm,m ∈ N, then ‖Bq −Br‖ = 2(m− 1)/m.