A Small Intervals Theorem for Subharmonic Functions

TL;DR

This paper establishes a new upper estimate for the integral of a subharmonic function over small intervals, extending classical theorems and providing uniform bounds independent of the specific function.

Contribution

It develops a novel upper estimate for subharmonic functions over small intervals, generalizing classical theorems and ensuring constants are absolute and independent of the function.

Findings

Provides an upper bound for the integral of |u| over subsets of E.

Extends Nevanlinna's classical theorem to smaller intervals.

Offers uniform estimates with absolute constants.

Abstract

Let $\mathbb C$ be the complex plane, $E$ be a measurable subset in a segment $[0, R]$ of the positive semiaxis $\mathbb R^+$, $u\not\equiv -\infty$ be a subharmonic function on $\mathbb C$. The main result of this article is an upper estimate of the integral of the module $|u|$ over a subset of $E$ through the maximum of the function $u$ on a circle of radius $R$ centered at zero and a linear Lebesgue measure of subset $E$. Our result develops one of the classical theorems of R. Nevanlinna in the case of $E=[0, R]$ and versions of so-called Small Arcs Lemma by Edrei-Fuchs for small intervals on $\mathbb R^+$ from the works of A.F. Grishin, M.L. Sodin, T.I. Malyutina. Our obtained estimate is uniform in the sense that the constants in the estimates are absolute and do not depend on the subharmonic function under the semi-normalization $u(0)\geq 0$.