I have trouble proving the Subordination principle for the Harnack distance.
If $G\subseteq \mathbb{C}$ is open, the Hanack distance $\tau_G(z,w)$ is defined as the smallest number such that $$h(w)\tau_G(z,w)^{-1}\leq h(z)\leq \tau_G(z,w)h(w)$$ for every positive harmonic function $h$ on $G$.
Now the subordination principle states: Let $f:G_1\to G_2$ be a holomorphic mapping between two domains. Then $\tau_{G_{2}}(f(z),f(w))\leq \tau_{G_{1}}(z,w)$ with equality if $f$ is conformal.
It's true that $$h(f(w))\tau_{G_{1}}(z,w)^{-1}\leq h(f(z))\leq \tau_{G_{1}}(z,w)h(f(w))$$ but I can't seem to figure out to show the last part.
Also, why does equality hold if $f$ is conform?