= Blaschke factorization of a bounded holomorphic function
{c}
{title2=$f=Bg,\quad \|g\|_\infty\le\|f\|_\infty$}
Every nonzero bounded <holomorphic function> on the <unit disc> has this factorization, with $B$ the <Blaschke product> of its zeros and $g$ zero-free and <holomorphic>. The <Blaschke condition> follows from <Jensen's formula>. Dividing by any finite partial zero product gives a <holomorphic function>; the <maximum modulus principle> and the finite product's boundary modulus one bound that quotient by $\|f\|_\infty$. <Locally uniform convergence> of the partial products gives the same bound for $g$, with removable extensions at their zeros. This factorization alone does not make $g$ constant or a full inner-outer factorization.
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