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 \frac{\rho}{2}\big(1-|f(0)|^{2}\big)+\sqrt{\frac{\rho^{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.

翻译：设 $A$ 为一个 $\rho$-压缩算子，$f$ 为一个将闭单位圆盘映射到自身的有理函数。通过建立 $\rho$-压缩算子的一个新刻画，我们证明了 \begin{align*} \big\|f(A)\big\|\leq \frac{\rho}{2}\big(1-|f(0)|^{2}\big)+\sqrt{\frac{\rho^{2}}{4}\big(1-|f(0)|^{2}\big){}^{2}+|f(0)|^{2}}. \end{align*} 我们进一步证明该界是尖锐的。这一结果改进了 Okubo--Ando 的估计，并且当 $\rho=2$ 时，与 Drury 的一个结论一致。