Let f be holomorphic on and inside a Jordan curve C and let

Let f be holomorphic on and inside a Jordan curve C and let f have maximum value M on C (such a number exists by the Extreme Value Theorem). Suppose that |f(z0)| = M for some z0 ins(C). Show that f is constant over ins(C).

Solution

By maximum modulus principle, if |f| attains the maximum value at the interior , it should be the constant function. Hence the result