On the growth of the integral means of subharmonic functions of order less than one

Abstract:

Let u be a subharmonic function of order $\lambda (0 < \lambda < 1)$, and let ${m_s}(r,u) = {\left \{ {(1/2\pi )\int _{ - \pi }^\pi {{{\left | {u(r{e^{i\theta }})} \right |}^s}} d\theta } \right \}^{1/s}}$. We compare the growth of ${m_s}(r,u)$ with that of the Riesz mass of u as measured by $N (r,u) = (1/2\pi )\int _{ - \pi }^\pi {u(r{e^{i\theta }})d\theta }$. A typical result of this paper states that the following inequality is sharp: \begin{equation} \tag {$\ast $} \underset {x\to \infty }{\lim \inf } \frac {{{m}_{s}}\left ( r, u \right )}{N\left ( r, u \right )} \leqslant {{m}_{s}}\left ( {{\psi }_{\lambda }} \right )\end{equation} where $\psi _\lambda (\theta ) = (\pi \lambda /\sin \lambda )\cos \lambda \theta$. The case $s = 1$ is due to Edrei and Fuchs, the case $s = 2$ is due to Miles and Shea and the case $s = \infty$ is due to Valiron.