If $\abs{f}\le M$ on the circle $\abs{z-z_0}=R$, then $\abs{f^{(n)}(z_0)}\le n!M/R^n$.Suggest a slogan
Let \(z_0\in\CC\) and \(R>0\), let \(f\) be holomorphic on an open set containing the closed disc \(\abs{z-z_0}\le R\), and let \(\gamma\) be the circle \(\abs{z-z_0}=R\). Then for every integer \(n\ge0\), \[ \abs{f^{(n)}(z_0)} \leq \frac{n!\, \sup_{z\in\gamma} \abs{f(z)}}{R^{n}}. \]
State the Cauchy inequalities for $f^{(n)}(z_0)$.
Backlinks
FT-REHJTCauchy inequalities for Taylor coefficientsvariant-of