On some characteristics of the growth of subharmonic functions

The connection between the growth of a function which is subharmonic in the plane and the growth of its associated Riesz measure is studied. The principal result (actually obtained in a more general form) is:
Theorem. {\it Suppose that the function $h(r)$ is differentiable on $(0,\infty)$, with $h'(x)>0$ and
$$ \lim_{x\to\infty}\frac{\ln x}{h(x)}=0,\qquad\lim_{x\to\infty}\frac{x\cdot h'(x)}{h(x)}=0. $$
Define
$$ \alpha_h(r)=\max_{1<\theta<\infty}\frac{\ln\theta}{h(\theta\cdot r)},\qquad\Delta_h=\varliminf_{r\to\infty}rh'(r)\alpha_h(r). $$
Suppose further that $\varphi(u)$ is a function which is subharmonic in $\mathbf R^2$, is of zero order, and has associated measure $\mu$. Then
\begin{gather*} \Delta_h\varlimsup_{r\to\infty}\frac{\mu(r)}{rh'(r)}\leqslant\varlimsup_{r\to\infty}\frac{M_\varphi(r)}{h(r)} \leqslant\varlimsup_{r\to\infty}\frac{\mu(r)}{rh'(r)},\\ \varliminf_{r\to\infty}\frac{M_\varphi(r)}{h(r)}\geqslant\varliminf_{r\to\infty}\frac{\mu(r)}{rh'(r)}, \end{gather*}
where
$$ \mu(r)=\mu(|z|\leqslant r),\qquadM_\varphi(r)\max\bigl\{0,\{\varphi(u):|u|=r\}\bigr\}. $$
If, in addition, $x\cdot h'(x)/h(x)$ is nonincreasing, then $\Delta_h\geqslant1/e$.}
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