Solve the below:
Q: Use the function f(z) = z to show that in Exercise the condition f(z) does not equal 0 anywhere in P is necessary in order to obtain the result of that exercise. That is, show that |f(z)| can reach its minimum value at an interior point when that minimum value is zero.
Exercise : Let a function f be continuous in a closed hounded 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 if (z)I 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. 50) to the function g(z) = 1/1(z).