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 (2104.11086v1)

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.