 Vanishing absolute logarithmic mean characterization of Blaschke products

For a holomorphic function on the unit disc, this boundary condition characterizes Blaschke products, including unimodular constants. The nondecreasing radial means of the nonnegative subharmonic function ${\log }^{+}|f|$ must vanish, so $|f|\le 1$. In the Blaschke factorization of a bounded holomorphic function $f=Bg$, both logarithmic means tend to zero. The mean value property for harmonic functions applied to $\log |g|$ gives $|g(0)|=1$, and the maximum modulus principle makes $g$ unimodular constant. The converse follows from the radial logarithmic mean of a Blaschke product. Omitting the outer absolute value changes the condition: $e^z$ has zero mean logarithmic modulus, but is not a Blaschke product.