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Proof Using the Maximum Modulus Principle

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Let f be continuous on a closed bounded region R and let it be analytic, and not constant throughout the interior of R. Assuming f(z) is not equal to 0 inside R (this assumption is easy since we may add a finitely large constant to f to avoid having it touch 0. Prove that |f(z)| has a minimum value m in R, which occurs on the boundary and never in the interior.

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In this solution we provide a proof using the maximum modulus principle.

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** Please see the attachment for the complete solution **

Since (please see the attached file) is analytic in the closed bounded region (please see the attached file), and (please see the attached file), then we can set (please see ...

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