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The zero-free quotient $g$ from 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 solution. It follows thatSince $|g|\le 1$ and $|g(0)|=1$, the maximum modulus principle makes $g$ a constant of modulus one. Absorb that constant into the unimodular factor of $B$. 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 $\log |B|\le 0$.