The zero set of a Hardy function satisfies the Blaschke condition

Reduction at the origin. By [L1] write $f(z)=z^{m}g(z)$ with $m\ge 0$, $g$ holomorphic on $\mathbb{D}$ and $g(0)\mathrm{\ne }0$; this $m$ is the finite order of the zero of $f$ at the origin, and $f$ has no other zeros at the origin. For $0<r<1$, ${\int }_{\mathbb{T}}\log \mathrm{\mid }f(r\zeta )\mathrm{\mid }dm(\zeta )=m\log r+{\int }_{\mathbb{T}}\log \mathrm{\mid }g(r\zeta )\mathrm{\mid }dm(\zeta )$ (for $m=0$ this is the identity; for $m\ge 1$ it holds because $\log \mathrm{\mid }{r}^m{\zeta }^m\mathrm{\mid }=m\log r$ is constant on the circle). Hence ${\mathrm{lim\hspace{0.17em}inf}}_{r\uparrow 1}{\int }_{\mathbb{T}}\log \mathrm{\mid }g(r\zeta )\mathrm{\mid }dm(\zeta )={\mathrm{lim\hspace{0.17em}inf}}_{r\uparrow 1}{\int }_{\mathbb{T}}\log \mathrm{\mid }f(r\zeta )\mathrm{\mid }dm(\zeta )<+\mathrm{\infty }$, because $m\log r\to 0$.

The logarithmic Jensen inequality at each radius. Let $F$ be holomorphic on a neighbourhood of the closed unit disc and not identically zero. Applying [L3] to $Y:=\mathrm{\mid }F{\mathrm{\mid }}^p$ with $0<p<\mathrm{\infty }$, and noting ${\int }_{\mathbb{T}}\mathrm{\mid }F{\mathrm{\mid }}^{p}dm=\mathrm{\parallel }F{\mathrm{\parallel }}_{L^p}^p<+\mathrm{\infty }$ because $F$ is continuous, gives ${\int }_{\mathbb{T}}\log \mathrm{\mid }F\mathrm{\mid }dm=\frac{1}{p}{\int }_{\mathbb{T}}\log \mathrm{\mid }F{\mathrm{\mid }}^{p}dm\le \frac{1}{p}\log {\int }_{\mathbb{T}}\mathrm{\mid }F{\mathrm{\mid }}^{p}dm=\log \mathrm{\parallel }F{\mathrm{\parallel }}_{L^p}\mathrm{.}$ For $p=\mathrm{\infty }$, $\log \mathrm{\mid }F\mathrm{\mid }\le \log \mathrm{\parallel }F{\mathrm{\parallel }}_{\mathrm{\infty }}$ pointwise, so ${\int }_{\mathbb{T}}\log \mathrm{\mid }F\mathrm{\mid }dm\le \log \mathrm{\parallel }F{\mathrm{\parallel }}_{\mathrm{\infty }}$.

Jensen's formula for $g$. Fix $0<r<1$ such that no $\mathrm{\mid }{a}_n\mathrm{\mid }$ equals $r$, and apply [L2] with $R=1$ to $F:=g(r\cdot )$, whose zeros in $\mathrm{\mid }z\mathrm{\mid }<1$ are the points $a_n\mathrm{/}r$ for those zeros $a_n$ of $f$ (equivalently of $g$) with $0<\mathrm{\mid }{a}_n\mathrm{\mid }<r$: $-{\sum }_{0<\mathrm{\mid }{a}_n\mathrm{\mid }<r}\log \frac{r}{\mathrm{\mid }{a}_n\mathrm{\mid }}=\log \mathrm{\mid }g(0)\mathrm{\mid }-{\int }_{\mathbb{T}}\log \mathrm{\mid }g(r\zeta )\mathrm{\mid }dm(\zeta )\mathrm{.}$

The $H^p$ clause. Let $f\in H^p(\mathbb{D})$. If $p=\mathrm{\infty }$, then for every $0<r<1$ step 1.2 applied to $f_r$ and [L4] give ${\int }_{\mathbb{T}}\log \mathrm{\mid }f(r\zeta )\mathrm{\mid }dm(\zeta )\le \log \mathrm{\parallel }{f}_r{\mathrm{\parallel }}_{\mathrm{\infty }}\le \log \mathrm{\parallel }f{\mathrm{\parallel }}_{H^{\mathrm{\infty }}}<+\mathrm{\infty }$, so the liminf hypothesis holds (when $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^{\mathrm{\infty }}}=0$ then $f\equiv 0$, excluded). If $0<p<\mathrm{\infty }$, the same steps give ${\int }_{\mathbb{T}}\log \mathrm{\mid }f(r\zeta )\mathrm{\mid }dm(\zeta )\le \log \mathrm{\parallel }{f}_r{\mathrm{\parallel }}_{L^p}\le \log \mathrm{\parallel }f{\mathrm{\parallel }}_{H^p}<+\mathrm{\infty }$.

Good radii and their limiting means. Jensen's formula in step 1.3, together with its boundary-zero limiting form in [L2], says that $M(r):=\int \log \mathrm{\mid }g(r\zeta )\mathrm{\mid }dm=\log \mathrm{\mid }g(0)\mathrm{\mid }+{\sum }_{0<\mathrm{\mid }{a}_n\mathrm{\mid }<r}\log (r\mathrm{/}\mathrm{\mid }{a}_n\mathrm{\mid })$ for every $0<r<1$; a zero on the radius contributes zero in that limiting identity. Thus $M(r)$ is nondecreasing. There are finitely many zero moduli in any closed subdisc, so a strictly increasing sequence of radii avoiding them and tending to $1$ can be chosen recursively, for instance from the rational radii in successive intervals tending to $1$. Monotonicity makes its means tend to ${\mathrm{lim\hspace{0.17em}inf}}_{r\uparrow 1}M(r)$. Along these radii, ${\sum }_{0<\mathrm{\mid }{a}_n\mathrm{\mid }<r_j}\log (r_j\mathrm{/}\mathrm{\mid }{a}_n\mathrm{\mid })=M(r_j)-\log \mathrm{\mid }g(0)\mathrm{\mid }$ by step 1.3.

The Blaschke condition. In the identity of step 2.2 the left-hand side $S_j:={\sum }_{0<\mathrm{\mid }{a}_n\mathrm{\mid }<r_j}\log (r_j\mathrm{/}\mathrm{\mid }{a}_n\mathrm{\mid })$ has nonnegative terms that increase with $j$ and eventually include every nonzero zero, so $S_j$ increases to $S:={\sum }_{n:a_n\mathrm{\ne }0}\log (1\mathrm{/}\mathrm{\mid }{a}_n\mathrm{\mid })\in [0,+\mathrm{\infty }]$. By the choice of the radii, the right-hand side converges to $L:={\mathrm{lim\hspace{0.17em}inf}}_{r\uparrow 1}{\int }_{\mathbb{T}}\log \mathrm{\mid }g(r\zeta )\mathrm{\mid }dm(\zeta )-\log \mathrm{\mid }g(0)\mathrm{\mid }$, and $L<+\mathrm{\infty }$ by step 1.1; since $S_j\ge 0$ for every $j$, $L\ge 0$ as well, so $L\in [0,+\mathrm{\infty })$. As limits of the same identity, $S=L<+\mathrm{\infty }$. Since $\log (1\mathrm{/}x)\ge 1-x$ for $0<x\le 1$, also ${\sum }_{n:a_n\mathrm{\ne }0}(1-\mathrm{\mid }{a}_n\mathrm{\mid })\le {\sum }_{n:a_n\mathrm{\ne }0}\log \frac{1}{\mathrm{\mid }{a}_n\mathrm{\mid }}<+\mathrm{\infty }$, and the zero at the origin contributes the finite amount $m$ to the second sum; hence ${\sum }_{n\ge 1}(1-\mathrm{\mid }{a}_n\mathrm{\mid })<+\mathrm{\infty }$.

Assembly. Step 1.1 produces the finite order $m$ of the zero at the origin and transfers the liminf hypothesis from $f$ to $g$; step 2.1 verifies the hypothesis for $H^p$ functions; steps 1.3–3.1 convert Jensen's formula for the dilated functions $g(r\cdot )$ into convergence of the zero sum $\sum \log (1\mathrm{/}\mathrm{\mid }{a}_n\mathrm{\mid })$ over the nonzero zeros, and then into the Blaschke condition $\sum (1-\mathrm{\mid }{a}_n\mathrm{\mid })<+\mathrm{\infty }$.