Let f (z) = u(x, y) + iv(x, y) be a function that is continuous on a closed bounded region R and analytic and not constant throughout the interior of R. Prove that the component function u(x, y) has a...

Let f (z) = u(x, y) + iv(x, y) be a function that is continuous on a closed bounded region R and analytic and not constant throughout the interior of R. Prove that the component function u(x, y) has a minimum value in R which occurs on the boundary of R and never in the interior. (See Exercise 3.)

Exercise 3

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 (Sec. 54) to the function g(z) = 1/f (z).