On some classes of subharmonic functions which have nonnegative harmonic majorants in a half-plane
The author studies functions \(u(z)\) subharmonic in the upper half-plane \(H\) which have nonnegative harmonic majorants there. The theory is less clean than with functions in the unit disk, because the natural exhaustion of \(H\) includes at each stage part of \(\partial H= \mathbb{R}\). The object is to make minimal assumptions on the boundary values of \(u\). The author introduces a function \(\Omega (x)\), \(-\infty< x< \infty\), with \(\Omega (x)> C(1+ x^2)^{-1}\) and the class \(S(\Omega)\) of subharmonic \(u\) in \(H\) with the standard condition \[ \liminf_{R\to \infty} R^{-1} \int_0^\pi u^+ (Re^{i \theta}) \sin (\theta) d\theta< \infty \tag \(*\) \] augmented by a somewhat intricate condition in terms of \(\Omega\) about the boundary values of \(u\) on \(\mathbb{R}\). He then shows that a representation analogous to that of Nevanlinna holds for functions of \(S(\Omega)\): \(u\) is a Green potential \(+\) the Poisson integral of a (signed) measure \(\mu\); \(\Omega\) plays a role in the exact form of \(\mu\). \{Reviewer's remarks. At the very end of \S 2, replace (1.20) by (1.21). The region \(G_{P,p}^{ (+)}\) in \S 2 is not a `segment', but the intersection of a disk with a half-plane\}.
scientific article; zbMATH DE number 795462