Representation of subharmonic functions in the half-ring and in the half-disk

K. G. Malyutin ; A. A. Naumova

Čebyševskij sbornik, Tome 24 (2023) no. 5, pp. 136-152.

Résumé

The work contains results in which representations of subharmonic functions on the most mentioned sets in a half-plane — in a half-ring and in a half-disk — are given. Classical results in this direction are, for example, the Nevanlinna, Poisson-Jensen and Shimizu-Ahlfors formulas of the representation of a meromorphic function in a closed disk and in a closed half-disk, as well as the Riesz-Martin theorem on the representation of subharmonic functions. In the works of T. Carleman (1933) and B. Ya. Levin (1941) for functions that are analytic and meromorphic in the closure of a half-ring and in the closure of a half-disk on the complex plane, formulas that relate the logarithm of the modulus of a function with the location of its zeros and poles were obtained. These formulas have found numerous applications in the theory of entire and meromorphic functions. Independently of each other, Jun-Iti Ito and A. F. Grishin (1968) extended the Levin and Carleman formulas to subharmonic functions in an open half-disk. Note, however, that Grishin's formulas using the Martin function, in our opinion, are more visual and convenient for practical use. In addition, A. F. Grishin formulated (without proof) a theorem on the representation of a subharmonic function in a semi-open half-ring. N. V. Govorov (1968) extended the Levin and Carleman formulas to analytic functions in a semi-closed half-disk and in a semi-closed half-ring. By the expression "semi-closed set" we mean a set on the complex plane, part of the boundary of which belongs to the set, and the rest of the boundary does not belong to it. In particular, by a semi-closed half-ring or a semi-closed half-disk in the upper half-plane of a complex variable we mean a half-ring or half-disk whose intersection of boundary with the real axis does not belong to the given set. In the article, we extend Grishin's formula to subharmonic functions in an open half-ring. We introduce the concept of full measure of a subharmonic function in an open half-ring, which generalizes the concept of full measure in the sense of Grishin. Due to this, the representation of the subharmonic function in the open half-ring, which is the simplest in form and with the least restrictions on the function, is obtained.

Détail

Keywords: Levin formula, Carleman formula, half-plane, half-ring, half-disk, subharmonic function, boundary measure, full measure, Riesz measure, singular measure.

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