Let \(f ( z ) = \sum _ { n = 0 } ^ { \infty } c _ { n } z ^ { n }\) be analytic and one-to-one in \(| z | < 1\) . For \(0 < r < 1\) , let \(D _ { r }\) be the disk
\(| z | < r\) . Show that the area of \(f ( D _ { r } )\) is finite and is given by
\[ S = \pi \sum _ { n = 1 } ^ { \infty } n | c _ { n } | ^ { 2 } r ^ { 2 n } . \]
(Note that in general the area of \(f ( D _ { 1 } )\) is infinite.)