Inner-outer factorization

Remarks

Proof idea: the zeros of $f$ satisfy the Blaschke condition, by Jensen's formula. Dividing by their Blaschke product $B$ gives a function $g = f/B$ in $H^p$ with no zeros and the same norm. The outer function $F$ with modulus $\abs{g^*}$ dominates $g$ in the sense that $\abs{g} \le \abs{F}$, because $\log \abs{g(z)}$ is at most the Poisson integral of $\log \abs{g^*}$ for $g \in H^p$. Then $g/F$ is bounded, has no zeros and has boundary modulus $1$, and such a function is a singular inner function $c S_\mu$.

Example: $f(z) = z (2 + z)$ is bounded, its inner factor is $z$ and its outer factor is $2 + z$.

The hypothesis $f \not\equiv 0$ is needed, since the zero function has no inner factor.

Going deeper

For a student

The factorization is the main structural tool for Hardy spaces. It reduces many questions to zero-free functions, for which one can take logarithms and roots: for example $f = B g$ with $g$ zero-free gives $g^{p/2} \in H^2$, which transfers results from $H^2$ to $H^p$. It underlies Beurling's theorem on shift-invariant subspaces, where outer functions are exactly the cyclic vectors of the shift on $H^2$. The same factorization holds in the Nevanlinna class if one allows quotients of singular inner functions.