A sharp bound for the functional calculus of ρ-contractions

Let A be a ρ-contraction and f a rational function mapping the closed unit disk into itself. With a new characterization of ρ-contractions we prove that
∥f(A)∥≤(ρ/2)(1−|f(0)|²)+√((ρ²/4)(1−|f(0)|²)²+|f(0)|²))
We further show that this bound is sharp. This refines an estimate by Okubo--Ando and, for ρ=2, is consistent with a result by Drury.