Cauchy's Inequality States that:
Let $f$ be analytic on $B(a, R)$ and let $r<R$. Then, for all $n \in \mathbb N$: $$\frac{|f^{(n)}(a)|}{n!}\leq \frac{\|f\|_{\partial B(0, r)}} {r^n}$$
The Open Map Theorem states that:
Let $f$ be analytic and non constant on a connected open set $A$. Then $f$ is an open map.
By using the Open Map Theorem, it's easy to prove the Maximum Module Principle, and, therefore, Cauchy's inequality for $n=0$. I was wondering: Since we can prove it for $n=0$, is it possible to prove Cauchy's Inequality by using the Open Map Theorem? Maybe by Induction?
I know some proofs of Cauchy's Inequality by using Integrals, However, I'm interested in knowing if there is a pfroof by using the open map Theorem.