Cauchy estimates for derivatives

Statement

Let $U$ be a domain in $\mathbb {C}$ and $f:U\to \mathbb {C}$ be a holomorphic function. Suppose there exists a nonnegative real constant $M$ such that:

$\left|f(z)\right|\leq M\ \forall \ z\in U$

Then, we have that for all $z\in U$ and for all $n\geq 0$:

$\left|f^{(n)}(z)\right|\leq {\frac {M(n!)}{dist(z,\partial U)^{n}}}$

Here $f^{(n)}(z)$ denotes the $n^{th}$ partial derivative of $f$, evaluated at the point $z\in \mathbb {C}$.

Facts used

These estimates are a direct consequence of the Cauchy integral formula for derivatives

Applications

• Bounded and entire implies constant: Any bounded entire function is constant
• Fundamental theorem of algebra