Hello, this was a problem on an exam:

From Cauchy's Integral Formula, we get Cauchy's Inequality $\displaystyle |f^{(n)}(z_0)| \leq \frac{n!M}{R^n}$.

Prove that equality holds if and only if $\displaystyle f(z)= \frac{aMz^n}{R^n}$ for some $\displaystyle a \in \mathbb{C}$ with $\displaystyle |a| = 1$.

Proving $\displaystyle (\Leftarrow)$ is easy, just take the modulus of the nth derivative of f.

How do we prove the other direction?

Thanks!