 Radial logarithmic mean of a Blaschke product

For $B(z)=\lambda z^m{\prod }_n{b}_{a_n}(z)$ with nonzero zeros $a_n$ counted with multiplicity, Jensen's formula gives the displayed identity. The negative logarithms of partial products increase, so the monotone convergence theorem permits termwise integration. The Blaschke condition implies ${\sum }_n-\log |a_n|<\infty$, which dominates the absolute summands. The dominated convergence theorem therefore gives a limit of zero as $r\uparrow 1$. Since $|B|\le 1$, the mean of $|\log |B||$ tends to zero as well.