Representation of subharmonic functions in a half-plane

Abstract

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 gamma(r) than finite order. The main result can be stated as follows. Let gamma(r) be a growth function such that either In gamma(r) is a convex function of In r or the lower order of gamma(r) is infinite. Then for each proper subharmonic function v of growth gamma(r) there exist an unbounded set R of positive numbers and a family {u(R) : R is an element of R} of proper subharmonic functions in the upper half-plane C+ such that