0$ there should exist a $\delta>0$ such that \[\int_A \log^+|f(rw)|dm(w)<\epsilon\] for all $A\subset T$ (the unit circle) with $m(A)<\delta$, and for all $r\in (0,1)$. The class $N$ is the usual Nevanlinna class, which can be viewed as the space of all functions on the unit disc which are quotients of bounded analytic functions. $N^+$ is the class of all functions on the unit disc which can be written as the quotient of a bounded analytic function with an outer function. See \cite{D} for details about these classes. It is left as an exercise for the reader to see that $N_*\subset N$, and, in fact, $N_*=N^+$. The following is then a corollary to Rudin's theorem: \begin{thm} Let $\phi\in N_*$. Then the set of points $w$ for which $\phi(z) - w$ has a non-trivial singular inner factor has logarithmic capacity zero. Conversely, given any (compact) set $E$ of logarithmic capacity zero, there is a bounded analytic function $\phi$ such that $\phi(z)-w$ has a non-trivial singular inner factor if and only if $w\in E$. \end{thm} The converse statement is well-known; see \cite{F}. Let $E$ be a compact set of capacity zero in $\Delta$. The covering map $F$ of the domain $\Delta\setminus E$ is an inner function since $E$ has capacity zero. For each $w\in E$, $\frac{F(z)-w}{1-\bar{w}F(z)}$ is a non-vanishing inner function and so is singular. Thus, since $1-\bar{w}F(z)$ is an outer function, $F(z)-w$ is a function with nontrivial singular inner factor for all $w$ in $E$. Sarason produced a different sort of extension of Frostman's result, which appears in a paper by R. Mortini \cite{M} as part of a constructive proof of the Beurling--Rudin theorem. He proved that for mutually prime inner functions $u$ and $v$ (by which we mean that $u$ and $v$ have no zero in common and that there is no singular inner function S with $u=Su_1$ and $v=Sv_1$ for inner functions $u_1$ and $v_1$), and $\rho>0$, the function $u(z)+\rho e^{it}v(z)$ has a trivial singular inner factor for almost all (with respect to Lebesgue measure) real $t$. Here we provide a generalization to the theorems of Frostman, Rudin, and Sarason to further answer the general question of when singular inner factors disappear. \begin{thm} Let $f,g \in H^p$, $0

0$) that \begin{align*} G(re^{i\theta})&=\int_E\log|f(re^{i\theta})-wg(re^{i\theta})|d\mu(w)\\ &\le \log(|O_f(re^{i\theta})|+|O_g(re^{i\theta})|)\\ &\le V_r(\theta) \end{align*} and it is clear that $V_r(\theta)\rightarrow V(\theta)$ pointwise. Since $O_f$ and $O_g$ are outer functions, we get $\int V_r(\theta)d\sigma\rightarrow\int V(\theta)d\sigma$ \cite[Theorem 2.10]{D}. To find the lower bound, we need the following lemma: \begin{lemma} There is a constant $K$ such that for all $z\in\Delta$, \begin{equation}\label{lemma1} G(z)\ge \log(\max\{|f(z)|, |g(z)|\})+K. \end{equation} \end{lemma} For the proof of the lemma, we break the unit disc up into two pieces, $A$ and $B$, where $A$ consists of those points $z$ where $|g(z)|\ge|f(z)|$, and $B$ those points where $|g(z)|<|f(z)|$. We will prove the lemma separately for points in $A$ and points in $B$. If $z\in A$, then $\log(\max\{|f(z)|, |g(z)|\}) = \log|g(z)|$. Recall that \[G(z)= \log|g(z)| + v\left(\frac{f(z)}{g(z)}\right)\] and $v(z)$, defined in equation \eqref{v}, is bounded below, so the term on the right above can be written as in \eqref{lemma1}. If $z\in B$, then we note that $|f(z)-wg(z)|>\frac{1}{2}|f(z)|$, so \[G(z)=\int_E\log|f(z)-wg(z)|d\mu(w)>\log(\frac{1}{2}|f(z)|).\] Also, for $z\in B$, $\log(\max\{|f(z)|, |g(z)|\}) = \log|f(z)|$, and again, \eqref{lemma1} can be satisfied. This completes the proof of the lemma. Now we will find the family $V_r(\theta)$ just as before. We note that \begin{equation}\label{lb1} \begin{split} \log(\max\{|f(re^{i\theta})|, |g(re^{i\theta})|\})\ge \log^-|&O_f(re^{i\theta})|+\log^-|O_g(re^{i\theta})|\\ &+ \log(\max\{|I_f(re^{i\theta})|,|I_g(re^{i\theta})|\}). \end{split} \end{equation} Take \begin{align}\label{V} V_r(\theta)&=\log^-|O_f(re^{i\theta})|+\log^-|O_g(re^{i\theta})| + \log(\max\{|I_f(re^{i\theta})|,|I_g(re^{i\theta})|\})\\ \intertext{and} V(\theta)&=\log^-|O_f(e^{i\theta})|+\log^-|O_g(e^{i\theta})|\notag \end{align} Putting together \eqref{lemma1}, \eqref{lb1}, and \eqref{V} gives us \[G(re^{i\theta})\ge V_r(\theta)+K.\] The sum of the first two terms in the definition of $V_r(\theta)$ in \eqref{V} clearly approaches $V(\theta)$ pointwise, and the third term approaches zero pointwise. Furthermore, \begin{align*} \int_T V_r(\theta)d\sigma = \int_T \log^-|&O_f(re^{i\theta})|+\log^-|O_g(re^{i\theta})|d\sigma(\theta)\\ &+ \int_T\log(\max\{|I_f(re^{i\theta})|,|I_g(re^{i\theta})|\})d\sigma(\theta), \end{align*} and the first term on the right, just as in the upper bound case, approaches $\int V(\theta)d\sigma(\theta)$. The second term approaches zero, which we can see from the %$\int\log(\max\{|I_f(re^{i\theta})|,|I_g(re^{i\theta})|\})d\sigma$, %has limit as %$r\rightarrow1$ of zero, by the following lemma by Sarason: \begin{lemma}\label{sarasonlemma} If $u_1$ and $u_2$ are inner functions without a common factor, then \[\lim_{r\rightarrow 1}\int_T\log(\max\{|u_1(re^{i\theta})|, |u_2(re^{i\theta})|\})d\sigma(\theta) = 0.\] \end{lemma} Sarason's proof of this lemma can be found in \cite{M}, but we include it here, with his permission, for completeness. The limit on the left side is the value at the origin of the least harmonic majorant in $\Delta$ of the subharmonic function $\max\{\log|u_1|,\log|u_2|\}$. So it remains to show that this least harmonic majorant is the constant function $0$. Let $h$ denote this least harmonic majorant. Then $\log|u_1|\le h\le0$. This implies that $h$ has radial limits $0$ almost everywhere on $T$. So, if $h$ is not identically zero, then $h$ is the Poisson integral of a negative singular measure on $T$. Hence $\varphi=e^{h+i\tilde h}$ is a singular inner function (here $\tilde h$ denotes the harmonic conjugate of $h$ in $\Delta$). Since $|u_1|\le|e^{h+i\tilde h}|$, the inner function $\varphi$ divides $u_1$. But $|u_2|\le|e^{h+i\tilde h}|$ implies that $\varphi$ also divides $u_2$, contradicting our assumption about $u_1$ and $u_2$. Thus $h\equiv 0$. This completes the proof of the lemma. Our conditions for convergence of the integrals in \eqref{mainconv} are satisfied, so we get \begin{align}\label{I} \lim_{r\rightarrow 1}I(r)&= \int\!\!\int\log{|f(e^{i\theta})-wg(e^{i\theta})|}\,d\sigma(\theta)\,d\mu(w).\\ \intertext{Next,} II(r)&=\int\!\!\int\log{|F_w(re^{i\theta})|}\,d\mu(w)\,d\sigma(\theta)\notag\\ &=\int\!\!\int\!\!\int\log{|f(e^{it})-wg(e^{it})|}P_r(\theta-t)\,d\sigma(t)d\sigma(\theta)d\mu(w)\nonumber\\ &= \int\!\!\int\log{|f(e^{it})-wg(e^{it})|}\,d\sigma(t)d\mu(w).\label{II} \end{align} We have $\int u(re^{i\theta})\,d\sigma(\theta) = I(r) - II(r)$. When \eqref{I} and \eqref{II} are used in this, we find $\lim_{r\rightarrow 1}\int u(re^{i\theta})\,d\sigma(\theta)=0$. As explained earlier, this finishes the proof that the set of $w$ for which $f(z)-wg(z)$ has a nontrivial singular factor has capacity zero. \end{proof} \begin{thebibliography}{99} \bibitem{CS} J. Caughran and A. Shields, \emph{Singular Inner Factors of Analytic Functions}, Michigan Math. J.,\textbf{16} (1969), 409-410. \bibitem{D} P. Duren, \emph{Theory of $H^p$ Spaces}, Academic Press, New York and London, 1970. \bibitem{F} S. Fisher, \emph{The Singular Set of a Bounded Analytic Function}, Michigan Math. J., \textbf{20} (1973), 257-261. \bibitem{G} J. Garnett, \emph{Bounded Analytic Functions}, Academic Press, New York, NY, 1981. \bibitem{H} K. Hoffman, \emph{Banach Spaces of Analytic Functions}, Prentice-Hall, Englewood Cliffs, NJ, 1962. \bibitem{L} N. S. Landkof, \emph{Foundations of Modern Potential Theory}, Springer-Verlag, Berlin--Heidelberg, 1982. \bibitem{M} R. Mortini, \emph{A Constructive Proof of the Beurling--Rudin Theorem}, preprint. \bibitem{R} W. Rudin, \emph{A Generalization of a Theorem of Frostman}, Math. Scand., \textbf{21} (1967), 136-143. \end{thebibliography} \end{document}