Working locally with affine coordinates $(y_{j})$ and complex coordinates

$$z_{j}={1\over 2\pi i}\log q_{j}=x_{j}+iy_{j},$$

we compute $\omega=2i\partial\bar{\partial}(K\circ f)={i\over 2}\sum{\partial^{2}K\over\partial y_{j}\partial y_{k}}dz_{j}\wedge d\bar{z}_{k}$ which is clearly positive. Furthermore, if $\Omega=dz_{1}\wedge\cdots\wedge dz_{n}$, then $\omega^{n}$ is proportional to $\Omega\wedge\bar{\Omega}$ if and only if $\det(\partial^{2}K/\partial y_{j}\partial y_{k})$ is constant. ∎