A sharp bound for the functional calculus of $\rho$-contractions

Let $A$ be a $\rho$-contraction and $f$ a rational function mapping the closed unit disk into itself. With a new characterization of $\rho$-contractions we prove that \begin{align*} \big|f(A)\big|\leq \fracρ{2}\big(1-|f(0)|^{2}\big)+\sqrt{\frac{ρ^{2}}{4}\big(1-|f(0)|^{2}\big){}^{2}+|f(0)|^{2}}. \end{align*} We further show that this bound is sharp. This refines an estimate by Okubo–Ando and, for $\rho =2$, is consistent with a result by Drury.