Representation of subharmonic functions in a half-plane

The theory of subharmonic functions of finite order is based to a considerable extent on integral formulae. In the present paper representations are obtained for subharmonic functions in the upper half-plane with more general growth γ(r) than finite order. The main result can be stated as follows. Let γ(r) be a growth function such that either lnγ(r) is a convex function of ln r or the lower order of γ(r) is infinite. Then for each proper subharmonic function v of growth γ(r) there exist an unbounded set R of positive numbers and a family (uR:R element of R) of proper subharmonic functions in the upper half-plane C+ such that 1) the full measures of the uR in the discs |z|≤R are equal to the full measure of the function v-uR→0 uniformly on compact subsets of C+ as R→∞, R element of R; 3) the function family {uR:R element of R} satisfies the growth constraints uniformly in R, that is, T(r,uR)≤Aγ(Br)/r, where A and B are constants and T(r, · ) is the growth characteristic. Bibliography: 16 titles.