minimum modulus principle concept and proof

Question

From the Wikipedia, under maximum modulus principle, it states that "By switching to the reciprocal, we can get the minimum modulus principle. It states that if f is holomorphic within a bounded domain D, continuous up to the boundary of D, and non-zero at all points, then |f(z)| takes its minimum value on the boundary of D."

How does non-zero at all points prove the minimum modulus principle such that it attains the minimum? Also from the context, it appears to use the reciprocal (1/f) but does not know how minimum modulus principle works.

Answer 1

If $f$ is nonzero and holomorphic within $D$, then the reciprocal $g = 1/f$ is also holomorphic on within $D$. One can thus apply the maximum modulus principle to $g$, which tells us that $\vert g \vert$ achieves its maximum on the boundary of $D$. Note that the maximum of $\vert g \vert$ is achieved at the point where $\vert f \vert$ achieves its minimum, so we obtain the minimum modulus principle.