Hyperbolic derivative density ($H_g(z)=2|g^{\prime }(z)|/(1-|g(z){|}^2)$)

For a nonconstant holomorphic map into the unit disc, the displayed density pulls the target hyperbolic metric back to the source Euclidean coordinate. It is not the operator norm between two hyperbolic tangent spaces, which additionally divides by the source metric density. Away from ramification points $\Delta \log H_g=H_g^2$; a ramification point of multiplicity $m$ adds $2\pi m$ times its point mass to the distributional Laplacian.

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.

For $g^{\prime }(0)\ne 0$, apply Jensen's formula with the smooth curvature term to the hyperbolic derivative density. Its distributional Laplacian is $H_g^2+2\pi \sum m_a{\delta }_a$, giving the displayed identity with ramification points defined by $g^{\prime }(a)=0$. If $g^{\prime }(z)=c{z}^m+\cdots$ at zero, use $m\log R$ as the origin counting term and replace the base constant by $\log {\lim }_{z\to 0}{H}_g(z)/|z{|}^m$. Without this regularization the basepoint-critical formula is undefined.