Area formula for a univalent power series ($\mathrm{Area}(f(r\mathbb{D}))=\pi {\sum }_{n\ge 1}n|c_n{|}^2{r}^{2n}$)

For an injective holomorphic function $f(z)={\sum }_{n\ge 0}{c}_n{z}^n$ on the unit disc, its real Jacobian determinant is $|f^{\prime }(z){|}^2$. Change of variables gives image area ${\int }_{|z|<r}|f^{\prime }(z){|}^{2}dA$. Angular orthogonality removes cross terms of the differentiated power series; integrating $2\pi n^2|c_n{|}^2{\rho }^{2n-1}$ radially gives the title formula. For $r<1$, uniform convergence on a slightly larger closed disk justifies termwise operations. Injectivity is essential for this to be image area rather than multiplicity of a parametrized surface integral.