$\int_a^b\frac{|\gamma'(t)|}{1-|\gamma(t)|^2}dt$ (from a calculation of Poincare distance)?

Question

This answer requires only 2 new definitions and elementary complex analysis.

I have been reading these notes titled Invariant Pseudodistances and Pseudometrics in Complex Analysis in Several Variables (PDF link via diva-portal.org). On page 13, section 1.4.3, they define the Poincare distance. Can you help me with the calculations where they compute $L_{{\beta}_\mathbb{D}}$

I know that the portion marked in yellow is a typo. But Can you tell me how $L_{{\beta}_\mathbb{D}}$ Is equal to the expression marked in red with a $dt$. And also why ${\beta}^i_\mathbb{D}(0, \phi_z(w))$ is equal to ${\beta}^i_\mathbb{D}(0, |\phi_z(w)|)$?

Answer 1

The superscripts $^3$ in the formula should all be $\ge$, which means that $L_{\beta_{\mathbb{D}}}(\gamma)$ is not equal, but greater or equal to the expression underlined in red. These inequalities are all pretty straightforward, e.g., the last one just follows from $|\gamma_{r_+}'(t)| \ge \gamma_{r_+}'(t)$ and $|\gamma_{r_+}(t)|^2 = \gamma_{r_+}(t)^2$. The fact that the Poincare distance between $0$ and $\zeta$ is the same as the one between $0$ and $|\zeta|$ for all $\zeta \in \mathbb{D}$ follows from the fact that it is invariant under rotations.