I've no idea what theorem can be useful to prove the following : Let $\displaystyle f$ be a continuous function in a closed and bounded region $\displaystyle R$, assume also that $\displaystyle f$ is analytic and not constant in the interior of $\displaystyle R$.

Assuming that $\displaystyle f(z)\neq 0$ in all $\displaystyle R$, prove that $\displaystyle |f(z)|$ has a minimum and that it is reached in the frontier of $\displaystyle R$ and never in the interior of $\displaystyle R$.