The Nevanlinna characteristic and integral inequalities with maximum radial characteristic for meromorphic functions and for the differences of subharmonic functions

Let $f$ be a meromorphic function on the complex plane $\mathbb C$ with Nevanlinna characteristic $T(r,f)$ and with maximal radial characteristic $\ln M(t,f)$, where $M(t,f)$ is the maximum of $|f|$ on the circle centered at zero and of radius $t$. РA series of known and widely used results make it possible to obtain upper estimates the integrals of $\ln M (t,f)$ over sets $E$ On the intervals $[0,r]$ in terms of $T(r,f)$ and the linear Lebesgue measure on $E$. In the paper, similar estimates are obtained for Lebesgue—Stieltjes of $\ln M(t,f)$ with respect to a monotone increasing function $m$, where the sets $E$ of nonconstancy for $m$ may be of fractal nature. It turns out to be possible to obtain nontrivial estimates in terms of the Hausdorff $h$-content and Hausdorff $h$-measure of $E$, and also in terms of their $d$-dimensional power versions with $d\in (0,1]$. All previously known estimates correspond to the extreme case of $d=1$ and an absolutely continuous function $m$ whose density belongs to $L^p$ with $p>1$. A substantial part of the exposition is presented at once for the differences of subharmonic or $\delta$-subharmonic functions on disks centered at zero, moreover, explicit estimational constants are found. The only restriction in the main theorem is that the modulus of continuity of $m$ must satisfy the Dini condition at zero, and this is essential, as is shown by a counterexample.