= Hyperbolic zero estimate for a Blaschke product
{title2=$|B(w)|\le\exp[-2\sum_n e^{-\rho(w,a_n)}]$}
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 $\log[(1+t)/(1-t)]\ge2t$. 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>.
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