4. Let a function f be continuous in a closed bounded region R, and let it be analytic and not constant throughout the interior of R. Assuming that f(z) does not equal 0 anywhere in R, prove that f(z)f has a minimum value n in R which occurs on the boundary of R and never in the interior. Do this by applying the corresponding result for maximum values (Sec. 50) to the function g(z) = 1/f(z).
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The minimum of a closed, continuous analytic function is investigated.