= Vanishing absolute logarithmic mean characterization of Blaschke products
{title2=$\int|\log|f(r\xi)||\,dm\to0\iff f\text{ is a Blaschke product}$}
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 $|f|\le1$. 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>.
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