Lemma. Let be analytic inside and on the boundary of the disk and satisfy for . If then for all in the disk.
Proof. Consider any circle . By the Cauchy Integral Formula we know that . For each on this circle with or . Assume that there is a on this circle for which . Then since is continuous there must be a whole arc of the circle on which for on . Also on the rest of the circle. Write .
From here how do we establish that , a contradiction? From the M-L Inequality, we know that . How do we get this to ?