Integrals of differences of subharmonic functions. I. An integral inequality with Nevanlinna characteristic and modulus of continuity of measure

TL;DR

This paper establishes new integral inequalities relating the difference of subharmonic functions to their Nevanlinna characteristic and measure continuity, with applications to meromorphic functions.

Contribution

It introduces novel integral inequalities connecting subharmonic function differences, Nevanlinna characteristic, and measure continuity, extending results to meromorphic functions.

Findings

Derived integral inequality involving Nevanlinna characteristic and measure modulus of continuity.

Extended the inequality framework to meromorphic functions.

Provided explicit bounds for integrals of meromorphic functions with respect to measures.

Abstract

We obtain new integral inequalities for the integrals of the difference of subharmonic functions in measure through their Nevanlinna characteristic and some functional characteristic of the measure. These results are new also for meromorphic functions. Let us illustrate the main result for the case of a meromorphic function. We denote by $D_z(r)$ a closed disc of radius $r$ centered at $z$ in the complex plane $\mathbb C$. Let $\mu\geq 0$ be a Borel measure on $D_0(r)$, and $M:=\mu(D_0(r))<+\infty$. The modulus of continuity of measure $\mu$ is the function $h(t):=\sup_{z\in \mathbb C}\mu(D_z(t)), \quad t\geq 0.$ Suppose that $\int_0\frac{h(t)}{t} dt<+\infty$. Let $f\neq 0$ be a meromorphic function on a neighbourhood of $D_0(R)$ of radius $R>r$ with $f(0)\in \mathbb C$ and the Nevanlinna characteristic $T_f$ . Then there is $$\int \ln^+|f| \,d\mu \leq 2\frac{R+r}{R-r}T_f(R)M\biggl(1+\int_0^{R+r}\frac{h(t)}{t} dt\biggr).$$