Hyperbolic characteristic of a disc map
= Hyperbolic characteristic of a disc map
{title2=$T_{\mathbb D}(R)=\frac1{2\pi}\int_{|z|<R}\log(R/|z|)H_g^2\,dA$}
This logarithmic area integral measures the accumulated pullback <hyperbolic metric> area of a <holomorphic> disc map. Its nonnegative kernel makes it increasing in $R$. <Schwarz-Pick theorem> bounds $H_g(z)$ by $2/(1-|z|^2)$, which controls its characteristic through a critical-point Jensen identity.