Boundary values and log-integrability of Nevanlinna-class functions

Apply [F1] to write $f=g\mathrm{/}h$ with $\mathrm{\mid }g\mathrm{\mid },\mathrm{\mid }h\mathrm{\mid }\le 1$ and $h$ zero-free. Both $g$ and $h$ are nonzero functions, since $f\equiv ̸0$. By [F2] they have finite nontangential limits $g^{\ast },h^{\ast }$ almost everywhere. We must prove $h^{\ast }\mathrm{\ne }0$ and the integrability of both boundary logarithms before dividing.

Let $q$ be any bounded nonzero holomorphic disc function, with bound $M>0$. By [F3], its origin order $m$ is finite; the quotient $q_0=q\mathrm{/}{z}^m$ off zero extends holomorphically to the disc with $q_0(0)\mathrm{\ne }0$. Its zeros on each closed annulus compactly inside the disc are finite: all open neighborhoods containing at most one zero cover the annulus, by continuity at nonzeros and isolatedness at zeros. Compactness in [F3] gives a finite subcover, bounding the number of zeros by its finite size. For every $j\ge 2$, choose $r_j\in (1-1\mathrm{/}j,1-1\mathrm{/}(j+1))$ whose circle contains no zero; only finitely many radii in this interval are forbidden. Countable choice supplies this sequence. Jensen applied to $q_0$ gives $\int \log \mathrm{\mid }q(r_j\zeta )\mathrm{\mid }dm=m\log r_j+\int \log \mathrm{\mid }{q}_0(r_j\zeta )\mathrm{\mid }dm\ge m\log (1\mathrm{/}2)+\log \mathrm{\mid }{q}_0(0)\mathrm{\mid }=:c>-\mathrm{\infty }\mathrm{.}$ Meanwhile ${\log }^{+}\mathrm{\mid }q(r_j\zeta )\mathrm{\mid }\le {\log }^{+}M=:C$. Consequently $\int {\log }^{-}\mathrm{\mid }q(r_j\zeta )\mathrm{\mid }dm\le C-c\mathrm{.}$ The nontangential limits $q^{\ast }$ of [F2] include radial limits; Fatou [F4] applied to the negative parts, with value $+\mathrm{\infty }$ at a zero boundary limit, gives $\int {\log }^{-}\mathrm{\mid }{q}^{\ast }\mathrm{\mid }dm\le C-c<\mathrm{\infty }$. The positive part is bounded by $C$. Thus $\log \mathrm{\mid }{q}^{\ast }\mathrm{\mid }\in L^1$ and $q^{\ast }\mathrm{\ne }0$ almost everywhere. This also covers $m=0$, a zero-free q and a nonzero constant.

Apply step 1.2 separately to $g$ and $h$. On the common full-measure set where their finite nonzero boundary limits exist, the quotient has finite nontangential limit $f^{\ast }=g^{\ast }\mathrm{/}{h}^{\ast }$. It is nonzero there, and $\log \mathrm{\mid }{f}^{\ast }\mathrm{\mid }=\log \mathrm{\mid }{g}^{\ast }\mathrm{\mid }-\log \mathrm{\mid }{h}^{\ast }\mathrm{\mid }\in L^1,$ by the triangle inequality for the two integrable logarithms. This identifies the complex limit directly, without inferring it from a modulus limit.

Step 2.1 proves the entire assertion. Only countable choice has been used: in the bounded-holomorphic boundary supplier [F2] and the sequence of good Jensen radii in step 1.2. No general positive-harmonic measure representation or full-AC decomposition is needed. The excluded zero function nevertheless has the obvious zero boundary function, a convention used by the class definitions without assigning it an integrable logarithm.