Cauchy's estimates | Complex Analysis | Mathematics

Theorem 3.1 (Cauchy’s estimates). Let \(f\) be analytic inside and on the circle \(C_R \colon \abs {z - z_0} = R\), with \(\abs {f(z)} \leq M\) on \(C_R\). Then

\[ \abs {f^{(n)}(z_0)} \leq \frac {n! \, M}{R^n} \qquad (n = 0, 1, 2, \dots ). \]

Source: [1, ch. 4]

Proof. On \(C_R\) the integrand in Theorem 2.2 satisfies \(\abs {f(z) / (z - z_0)^{n+1}} \leq M / R^{n+1}\), and \(C_R\) has length \(2\pi R\). By Theorem 1.1,

\[ \abs {f^{(n)}(z_0)} \leq \frac {n!}{2\pi } \cdot \frac {M}{R^{n+1}} \cdot 2\pi R = \frac {n! \, M}{R^n}. \qedhere \]