On some general classes of subharmonic functions (Q1896854)

The author develops representations for subharmonic functions of generalized bounded type in a half-plane, which here is \(\{\text{Im } z< 0\}\). They depend on a parameter \(\alpha> -1\); the classical (Nevanlinna) case is \(\alpha=0\). Following earlier work [Sov. Math., Dokl. 20, 607-610 (1979); translation from Dokl. Akad. Nauk SSSR 246, No. 6, 1295-1298 (1979; Zbl 0429.30028)]\ the author introduces a Blaschke-type product \(b_\alpha (z, \zeta)\) which has the key properties of the classical Blaschke product \((\alpha =0)\). Then if \(\nu\) is a Borel measure with \(\iint |\text{Im } z|^{1+ \alpha} dx dy< \infty\), a Green potential is constructed using the \(b_\alpha\). Using this, it is possible to characterize functions of certain classes \(s_\alpha\) and \(S_\alpha\); the definition of \(s_\alpha\) cannot be reproduced here, but \(S_\alpha\) \((0\leq \alpha<+ \infty)\) is the class of those functions subharmonic in the lower half-plane \(\{z: \text{Im } z<0\}\) which are representable as the difference of two functions of \(s_\alpha\). The methods involve (Weyl) fractional integration (used in the above reference also) and its inverse. The author observes that his methods shed insight on the Blaschke-type factorization of \textit{M. M. Dzhrbashyan} [Integral transforms and representations of functions in the complex domain (in Russian) (Moscow, Nauka, 1966; Zbl 0154.377)]. \{Remark. In some of these representations, it is assumed that the measure \(\nu\) vanishes in a neighborhood of \(\infty\)\}.