Let $f$ be a non-conformal analytic function from the unit disk to itself. Show that $f$ is strictly contracting with respect to the hyperbolic metric on any subdisk $\{|z|\le r\}$, $0<r<1$. Specifically, show that
$$\rho(f(z_0),f(z_1))\le c\rho(z_0,z_1), \quad |z_0|,|z_1|\le r, c < 1$$
where $\rho(z_0,z_1)$ is the hyperbolic distance from $z_0$ to $z_1$.
I made some progress, but I got stuck. Here's my progress:
Let $\gamma$ be the geodesic from $z_0$ to $z_1$. Then $$ \begin{align} \rho(f(z_0),f(z_1)) &\le 2\int_{f\circ\gamma} \frac{|dw|}{1-|w|^2}\\ &= 2\int_{\gamma} \frac{|f'(z)||dz|}{1-|f(z)|^2}\\ \end{align} $$
I thought that maybe I should make a change of variables and use Pick's theorem. However, this didn't get me anywhere. Hints would be appreciated.