The Bergman metric on the unit disk $D$ is defined as: $$\beta(z,w)=\frac{1}{2} log \frac{1+\rho(z,w)}{1-\rho(z,w)}$$ where $$\rho(z,w)=\left|\dfrac{z-w}{1-\bar wz}\right|.$$ is the pseudo-hyperbolic distance on the unit disk $D$.
I need to prove the triangle inequality for the Bergman metric, I think the easiest way is to assume that $\rho(z,w)$ is a metric and therefore the triangle inequality for the bergman metric is inherited from $\rho(z,w)$, but I really don't know how this could be written correctly or if I'm right, could you give me a idea how to prove the triangular inequality for the bergman metric?.
Thanks