Real and Complex Analysis (Rudin)

260 REAL AND COMPLEX ANAL YSlS
for ZEn, and put h e (1) = hA2r - 1) = o. If K is the union of E and the
bounded components of the complement of E, then K is compact, he is continuous
on K, holomorphic in the interior of K, and (5) implies that I he I < E
on the boundary of K. Since the construction of E shows that r E K, the
maximum modulus theorem implies that I hc