I was working on a question very similar to this one: Expressing the area of the image of a holomorphic function by the coefficients of its expansion
Clearly the key lies in the formula $\iint_{f(D)}dxdy=\iint_D\operatorname{Jac}(f)\,dx\,dy,$ which turns out to be $\iint_D|f'|^2\,dx\,dy$ for holomorphic functions (using Cauchy-Riemann equations). But it has been pointed out that this is only true for one-to-one functions. Intuitively this makes sense to me since the integral would sum the same area more than once if it were not one-to-one. But then...
Is it correct to generalize this formula to functions which are not one-to-one by saying $\iint_{f(D)}dxdy\leq \iint_D|f'|^2\,dx\,dy$ with equality when $f$ is one-to-one? Would equality also hold if the set of points in the image which have more than one pre-image is non-empty but has measure zero? Or can stranger things happen that I have not considered?
If my question makes more sense with $\operatorname{Jac}(f)$ in place of $|f'|^2$, please let me know, and feel free to use the former instead.
Thanks.