Cauchy's estimates

Statement

Let $U \subseteq \mathbb{C}$ be an open set and let $f : U \to \mathbb{C}$ be holomorphic on $U$. Let $c \in \mathbb{C}$ and $r > 0$ be such that the closed disc $\{z : |z - c| \le r\}$ is contained in $U$. Let $M \ge 0$ be a real number with $|f(z)| \le M$ for each $z$ with $|z - c| = r$. Then for each integer $n \ge 0$, $$|f^{(n)}(c)| \le \frac{n! \, M}{r^n}.$$

Remarks

Proof: apply the estimation lemma to the formula for the derivatives with $a = c$. The circle has length $2 \pi r$, and $|f(z) / (z - c)^{n+1}| \le M / r^{n+1}$ on it. For $n = 1$, the estimate is $|f'(c)| \le M / r$. If $f(z) = \sum a_n (z - c)^n$ is the Taylor series, the estimate is $|a_n| \le M / r^n$.

Explanations

L1 human

If a holomorphic function is small on a circle, then its derivatives at the center cannot be large. A larger circle with the same bound $M$ gives a better bound for the derivatives. This is the idea of Liouville's theorem.