Supremum and infimum of subharmonic functions of order between 1 and 2

For functions u, subharmonic in the plane, letand let N(r,u) be the integrated counting function. Suppose that $\mathcal{N}\colon[0,\infty)\rightarrow\mathbb{R}$ is a non-negative non-decreasing convex function of log r for which $\mathcal{N}(r)=0$ for all small r and $\limsup_{r\to\infty}\log\mathcal{N}(r)/\4\log r=\rho$, where 1 < ρ < 2, and defineA sharp upper bound is obtained for $\liminf_{r\to\infty}\mathcal{B}(r,\mathcal{N})/\mathcal{N}(r)$ and a sharp lower bound is obtained for $\limsup_{r\to\infty}\mathcal{A}(r,\mathcal{N})/\mathcal{N}(r)$.