Hyperbolic zero estimate for a Blaschke product ($|B(w)|\le \exp [-2{\sum }_n{e}^{-\rho (w,a_n)}]$)

Use curvature-minus-one Hyperbolic distance in the Poincare disc. Each Blaschke factor has modulus $\tanh (\rho (w,a)/2)=(1-t)/(1+t)$, with $t=e^{-\rho (w,a)}$. For $0\le t<1$, differentiating shows $\log [(1+t)/(1-t)]\ge 2t$. Sum this logarithmic inequality over the zero factors and pass to the Blaschke product by locally uniform convergence. At a prescribed zero, its modulus vanishes and the bound is immediate. All sums count multiplicities.