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).

Nov 18, 2021
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