The growth of subharmonic functions in a semicircle

Subharmonic functions $v$ in an unbounded open semiring, the growth of which is determined by the positive, continuous, increasing and unbounded function $\gamma(r)$, defined on $[0;\infty)$ (the growth function) are considered in the paper. Space of subharmonic functions of finite $\gamma$-type are denoted as $S(R, \gamma)$. In terms of Fourier coefficients, the criterion for belonging of a subharmonic function to the space $S(R, \gamma)$ is obtained. The paper contains some of the results by A.A. Kondratyuk, K.G. Malyutina, B.N. Khabibullina et al. 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 receiving the criteria for belonging of subharmonic function to a given class.