Cauchy inequalities for Taylor coefficients

If $\abs{f}\le M$ on the circle $\abs{z-z_0}=R$, then the $n$th Taylor coefficient of $f$ at $z_0$ has modulus at most $M/R^n$.Suggest a slogan

Let $z_0\in\CC$ and $R>0$, and let $f$ be holomorphic on an open set containing the closed disc $\abs{z-z_0}\le R$. Then for every integer $n\ge0$, \[ \abs{f^{(n)} (z_0) \over n!} \leq R^{-n} \sup_{\abs{z-z_0}=R} \abs{f(z)}. \]

State the Cauchy inequalities in the form bounding $\abs{f^{(n)}(z_0)/n!}$.