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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.

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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.

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