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 $\gamma(r)$ than finite order. The main result can be stated as follows. Let $\gamma(r)$ be a growth function such that either $\ln\gamma(r)$ is a convex function of $\ln r$ 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 $\mathbf R$ of positive numbers and a family $\{u_R:R\in\mathbf R\}$ of proper subharmonic functions in the upper half-plane $\mathbb{C}_+$ such that
1) the full measures of the $u_R$ in the discs $|z|\leqslant R$ are equal to the full measure of the function $v$;
2) $v-u_R\rightrightarrows0$ uniformly on compact subsets of $\mathbb{C}_+$ as $R\to\infty$, $R\in\mathbf R$;
3) the function family $\{u_R:R\in\mathbf R\}$ satisfies the growth constraints uniformly in $R$, that is, $T(r,u_R)\leqslant A\gamma(Br)/r$, where $A$ and $B$ are constants and $T(r,\,\cdot\,)$ is the growth characteristic.
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