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## Homework Statement

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

## Homework Equations

Corollary. Suppose that a function f is continuous on a closed bounded region R and that it is analytic and not constant in the interior of R. Then the maximum value of |f(z)| in R, which is always reached, occurs smoewhere on the boundary of R and never in the interior.

## The Attempt at a Solution

Suppose that g = 1/f is continuous on a closed bounded region R and that it is analytic and not constant in the interior of R. By Corollary, |g(z)| has a maximum value, and

when |g(z)| has the maximum value, |1/g(z)|=|f(z)| has the minimum value. Complete.

Is it right approach?