On some characteristics of the growth of subharmonic functions

A. V. Bratishchev ; Yu. F. Korobeinik

Sbornik. Mathematics, Tome 34 (1978) no. 5, pp. 603-626

Résumé

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 $\Delta_h\geqslant1/e$.} Bibliography: 12 titles.