Growth of Subharmonic Functions in a Semicircle

In this paper, subharmonic functions $${v}$$ in an unbounded open semiring, the growth of which is determined by the positive, continuous, increasing, and unbounded function γ(r) defined on [0, ∞) (the growth function), have been considered. The spaces of subharmonic functions of finite γ-type are denoted as S(R, γ). In terms of Fourier coefficients, the criterion for belonging of a subharmonic function to the S(R, γ) space has been obtained. In this paper, some of the results by A.A. Kondratyuk, K.G. Malyutin, B.N. Khabibullin, et al. have been extended to the functions defined in unbounded semiring. Transition to an unbounded semiring causes certain difficulties associated with complex behavior functions in a neighborhood of the boundary. Difference from the plane case appears already when obtaining the criteria for belonging of subharmonic function to a specified class.