= Solution
The zero-free quotient $g$ from <2/b/solution> has a <holomorphic logarithm>, so $\log|g|$ is <harmonic>. Its radial mean is the constant $\log|g(0)|$. Subtract the radial logarithmic means of $f$ and $B$. The first tends to zero by the absolute logarithmic hypothesis; the second tends to zero by <2/solution>. It follows that
$$
\log|g(0)|=\int\log|f(re^{i\theta})|\,dm-\int\log|B(re^{i\theta})|\,dm=0.
$$
Since $|g|\le1$ and $|g(0)|=1$, the <maximum modulus principle> makes $g$ a constant of modulus one. Absorb that constant into the unimodular factor of $B$. \b[Therefore $f$ itself is a <Blaschke product>.] This proves the <vanishing absolute logarithmic mean characterization of Blaschke products>; conversely a <Blaschke product> satisfies the absolute version too, since $\log|B|\le0$.
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