On the addition of the indicators of entire and subharmonic functions of several variables

In this article a necessary and sufficient criterion is derived for a subharmonic function $u(x)$ defined in $\mathbf R^p$ and having proximate order $\rho(t)$ to belong to the class of functions of completely regular growth. The criterion is that for any subharmonic function $v(x)$ with the same proximate order the sum of the regularized indicators of $u(x)$ and $v(x)$ be equal to the regularized indicator of the sum $u(x)+v(x)$. If the dimension of the space is $p=2l$ then it suffices to consider functions $v(x)$ of the type $\ln|f(z)|$, where $f(z)$ is an entire function on $\mathbf C^l$.
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