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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

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Let f be holomorphic on and inside a Jordan curve C and let f have m.pdf

  • 1. 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