1. ## Complex Analysis

1. If $f$ is analytic in a closed bounded region $G$ and $f(z) \neq 0$ in $G$, show
that $|f|$ assumes its minimum value on the boundary of $G$.
Hint: consider $\frac{1}{f}$.

2. Use Problem 1 to prove the Fundamental Theorem of Algebra.

2. Originally Posted by chiph588@
1. If $f$ is analytic in a closed bounded region $G$ and $f(z) \neq 0$ in $G$, show
that $|f|$ assumes its minimum value on the boundary of $G$.
Hint: consider $\frac{1}{f}$.
The minimum value of $|f|$ is the maximum value of $1/|f|$.
But $1/|f|$ assumes it maximum on the boundary by the maximum modulos principle.

3. Thanks!