Growth of subharmonic functions along line and distribution of their Riesz measures

Abstract

Let $u\not\equiv -\infty$ and $M\not\equiv -\infty$ be two subharmonic functions on a complex plane $\mathbb C$ with Riesz measures $\nu_u$ and $\mu_M$, respectively, such that $u(z)\leq O(|z|)$ and $M(z)\leq O(|z|)$ as $z\to \infty$, and $q$ is some positive continuous function on a real axis $\mathbb{R}$, and ${\rm mes}$ is a linear Lebesgue measure on $\mathbb{R}$. We assume that the following condition for the growth of function $u$ along the imaginary axis $i\mathbb{R}$ of the form $$ u(iy)\leq \frac{1}{2\pi}\int\limits_0^{2\pi}M\bigl(iy+q(y)e^{i\theta}\bigr)\,{\rm d}\theta +q(y) \quad\text{for all } y\in \mathbb{R}\setminus E, $$ where $E\subset \mathbb{R}$ is some small set, for instance, ${\rm mes}\bigl(E\cap [-r,r]\bigr)\leq q(r)$ as $r\geq 0$. Under such restrictions for the function $u$ it is natural to expect that the Riesz measure $\nu_u$ is in some sense majorized by the Riesz measure $\mu_M$ of the function $M$ or by integral characteristics of the function $M$. We provide a rigorous quantitative form of such majorizing. The need in such estimates arises naturally in the theory of entire functions in its applications to the completeness issues of exponential systems, analytic continuation, etc. Our results are formulated in terms of special logarithmic characteristics of measures $\nu_u$ and $\mu_M$ arisen earlier in classical works by P. Malliavin, L.A. Rubel and other for sequences of points and also in terms of special logarithmic characteristics of the behavior of the function $M$ along the imaginary axis and of the function $q$ along the real axis. The obtained results are new also for distribution of the zeroes of entire functions of exponential type under restrictions for the growth of such function along a line. The latter is demonstrated by a new uniqueness theorem for entire functions of exponential type employing so-called logarithmic block-densities of the distribution of the points on the complex plane.