Univalent functions on a punctured plane are Möbius transformations

Let \(f ( z )\) be an analytic function on \(\mathbb { C } \backslash \{ z _ { 0 } \}\) , where \(z _ { \mathrm { 0 } }\) is a fixed point. Assume that \(f ( z )\) is bijective from \(\mathbb { C } \backslash \{ z _ { 0 } \}\) onto its image, and that \(f ( z )\) is bounded outside \(D _ { r } ( z _ { 0 } )\) , where r is some fixed positive number. Show that there exist \(a , b , c , d \in \mathbb { C }\) with \(a d - b c \neq 0 , c \neq 0\) such that \(f ( z ) = { \frac { a z + b } { c z + d } } .\)