The idea of the proof is to look at the area uncovered by the image of f. Define for r\in(0,1) :\gamma_r(\theta):=f(r\,e^{-i\theta}),\qquad \theta\in[0,2\pi]. Then \gamma_r is a simple closed curve in the plane. Let D_r denote the unique bounded connected component of \mathbb C\setminus\gamma_r([0,2\pi]). The existence and uniqueness of D_r follows from
Jordan's curve theorem. If D is a domain in the plane whose boundary is a
smooth simple closed curve \gamma, then : \mathrm{area}(D)=\int_\gamma x\,dy=-\int_\gamma y\,dx\,, provided that \gamma is positively
oriented around D. This follows easily, for example, from
Green's theorem. As we will soon see, \gamma_r is positively oriented around D_r (and that is the reason for the minus sign in the definition of \gamma_r). After applying the
chain rule and the formula for \gamma_r, the above expressions for the area give : \mathrm{area}(D_r)= \int_0^{2\pi} \Re\bigl(f(r e^{-i\theta})\bigr)\,\Im\bigl(-i\,r\,e^{-i\theta}\,f'(r e^{-i\theta})\bigr)\,d\theta = -\int_0^{2\pi} \Im\bigl(f(r e^{-i\theta})\bigr)\,\Re\bigl(-i\,r\,e^{-i\theta}\,f'(r e^{-i\theta})\bigr)d\theta. Therefore, the area of D_r also equals to the average of the two expressions on the right hand side. After simplification, this yields : \mathrm{area}(D_r) = -\frac 12\, \Re\int_0^{2\pi}f(r\,e^{-i\theta})\,\overline{r\,e^{-i\theta}\,f'(r\,e^{-i\theta})}\,d\theta, where \overline z denotes
complex conjugation. We set a_{-1}=1 and use the power series expansion for f, to get : \mathrm{area}(D_r) = -\frac 12\, \Re\int_0^{2\pi} \sum_{n=-1}^\infty \sum_{m=-1}^\infty m\,r^{n+m}\,a_n\,\overline{a_m}\,e^{i\,(m-n)\,\theta}\,d\theta\,. (Since \int_0^{2\pi} \sum_{n=-1}^\infty\sum_{m=-1}^\infty m\,r^{n+m}\,|a_n|\,|a_m|\,d\theta the rearrangement of the terms is justified.) Now note that \int_0^{2\pi} e^{i\,(m-n)\,\theta}\,d\theta is 2\pi if n= m and is zero otherwise. Therefore, we get : \mathrm{area}(D_r)= -\pi\sum_{n=-1}^\infty n\,r^{2n}\,|a_n|^2. The area of D_r is clearly positive. Therefore, the right hand side is positive. Since a_{-1}=1, by letting r\to1, the theorem now follows. It only remains to justify the claim that \gamma_r is positively oriented around D_r. Let r' satisfy r, and set z_0=f(r'), say. For very small s>0, we may write the expression for the
winding number of \gamma_s around z_0, and verify that it is equal to 1. Since, \gamma_t does not pass through z_0 when t\ne r' (as f is injective), the invariance of the winding number under homotopy in the complement of z_0 implies that the winding number of \gamma_r around z_0 is also 1. This implies that z_0\in D_r and that \gamma_r is positively oriented around D_r, as required. ==Uses==