Absolute values and real parts for functions in the Smirnov class (Q1433182)

The Nevanlinna class \(N\) consists of all functions \(f\) holomorphic in the unit disc \(\mathbb D\) for which \(\sup_{0\leq r<1}\int^{2\pi}_0\log^+| f(re^{i\theta})| d\theta<\infty\). The Smirnov class \(N_+\) is the space of those \(f\in N\) such that \(\lim_{r\to 1}\int^{2\pi}_0\log^+ | f(re^{i\theta})| d\theta=\int^{2\pi}_0\log^+| f(e^{i\theta}) | d\theta\), where \(f(e^{i\theta})=\lim_{r\to 1}f(re^{i\theta})\) is the almost everywhere boundary value function on \(\partial \mathbb D\). For \(0<p\leq\infty\), the Hardy space \(H^p\) consists of those \(f\in N_+\) whose boundary value function belongs to \(L^p(\partial \mathbb D)\). An inner function is an \(H^\infty \)-function with radial limits of absolute value \(1\) a.e. on \(\partial \mathbb D\). A function \(g\in N_+\) is said to be outer if it is not divisible in \(N_+\) by a non-constant inner function. A stronger concept is that of a strongly outer function: A function \(g\in H^1\) is said to be strongly outer if the only functions \(f\in H^1\) such that \(f/g\geq 0\) on \(\partial \mathbb D\) are scalar multiples of \(g\). Strongly outer functions play an important role in many extremal problems in \(H^1\). They already appeared in the paper [Pac. J. Math. 8, 467--485 (1958; Zbl 0084.27503)] where \textit{K.~de~Leeuw} and \textit{W.~Rudin} proved that if \(f\in H^1\) and \(\| f\| _{H^1}=1\) then: (a) \(f\) is an extreme point of the unit sphere in \(H^1\) if and only if \(f\) is an outer function; and, (b) \(f\) is an exposed point of the unit sphere in \(H^1\) if and only if \(f\) is an strongly outer function. In this paper the author extends the notion of strongly outer function to \(N_+\) and, for an outer function \(g\in N_+\), he gives a necessary and sufficient condition for \(g\) to be strongly outer, and he shows that there exists an outer function \(G\in N_+\) such that \(| g| \leq{ Re}\,G\) on \(\partial \mathbb D\) if and only if \(g\in H^1\) or \(g\) is not strongly outer.