Question
Corollary. Suppose that a function f iscontinuous on a closed bounded region R and that it is analytic andnot constant in the interior of R. Then the maximum value of |f(z)|in R, which is always reached, occurs smoewhere on the boundary ofR and never in the interior. Problem. Let a function f be continuous on a closed bounded region R,and let it be analytic and not constant throughout the interior ofR. Assuming that f(z) != 0 anywhere in R, prove that |f(z)| has aminimum value m in R which occurs on the boundary of R and never inthe interior. Do this by applying the corresponding result formaximum values to the function g(z) = 1/f(z). Corollary. Suppose that a function f iscontinuous on a closed bounded region R and that it is analytic andnot constant in the interior of R. Then the maximum value of |f(z)|in R, which is always reached, occurs smoewhere on the boundary ofR and never in the interior. Problem. Let a function f be continuous on a closed bounded region R,and let it be analytic and not constant throughout the interior ofR. Assuming that f(z) != 0 anywhere in R, prove that |f(z)| has aminimum value m in R which occurs on the boundary of R and never inthe interior. Do this by applying the corresponding result formaximum values to the function g(z) = 1/f(z).
Explanation / Answer
let g be a nonconstant analytic function on a closed bounded region. then by the max. mod principle |g(z)| >=z for all z. Now take the reciprocal: 1/|g(z)|