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

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 $$\int \ln^+|f| \,d\mu \leq 2\frac{R+r}{R-r}T_f(R)M\biggl(1+\int_0^{R+r}\frac{h(t)}{t} dt\biggr).$$