Boundary control on a circle bounds every derivative at the center by $n!M/r^n$.Suggest a slogan
Let \(r>0\) and let \(f\) be holomorphic on an open set containing the closed disc \(\abs{z}\le r\). Then for every integer \(n\ge0\), \[ \abs{f^{(n)}(0)} \leq \frac{n!}{r^{n}} \sup_{\abs{z}=r}\abs{f(z)}. \]
State the Cauchy estimate for $\abs{f^{(n)}(0)}$.