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