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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

  • 1
    ALternatively, $\log|f(z)|$ is harmonic, if you know about harmonic functions.2012-04-29
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    Do you know anything about $R$?2012-04-29
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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.2012-04-29

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