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Let $f(z)$ be analytic and nonzero in a region R. Show that $|f(z)|$ has a minimum value in R that occurs on the boundary.

I think you should use the Maximum-Modulus Theorem for the function $1/f(z)$

The Maximum-Modulus Theorem

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ALternatively, $\log|f(z)|$ is harmonic, if you know about harmonic functions. – GEdgar Apr 29 '12 at 2:56
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Do you know anything about $R$? – Brett Frankel Apr 29 '12 at 2:59
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A region is an open set, so if $f$ nonzero in $R$, and $|f|$ has a minimum value in $R$, then by Strong Maximum-Modulus Theorem for $1/f$, $f$ must be constant. – jerrysciencemath Apr 29 '12 at 3:02

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