Meromorphic functions and differences of subharmonic functions in integrals and the difference characteristic of Nevanlinna. II. Explicit estimates of the integral of the radial maximum growth characteristic

TL;DR

This paper provides explicit, optimal estimates for the integral of the radial maximum growth characteristic of the difference of subharmonic functions, linking it to the Nevanlinna difference characteristic and the modulus of continuity.

Contribution

It offers explicit and optimal estimates for the integral of the radial maximum growth characteristic under specific continuity conditions, extending previous general bounds.

Findings

Estimates are explicit and depend on the modulus of continuity of the function m.

Conditions on the modulus of continuity include a differentiability constraint.

Results are optimal within the specified class of functions.

Abstract

Let $U\not\equiv \pm\infty$ be the difference of subharmonic functions, i.e., a $\delta$-subharmonic function, on a closed disc of radius $R$ centered at zero. In the preceding first part of our paper, we obtained general estimates for the integral of the positive part of the radial maximum growth characteristic ${\mathsf M}_U(t):=\sup\bigl\{U(z)\bigm| |z|=r\bigr\}$ over the increasing integration function $m$ on the segment $[0, r]$ via the Nevanlinna difference characteristic and the modulus of continuity of the function $m$. The second part of the work gives an explicit view for such estimates, provided that the modulus of continuity of the function $m$ does not exceed some differentiable function $h$ on the open interval $(0,r)$ with the only condition $\sup\limits_{t\in (0,r)}\dfrac{h(t)}{th'(t)}<+\infty$. This condition is satisfied by any power functions $t\mapsto t^d$ of degree $d>0$. The estimates are optimal in a certain sense.