Nevanlinna characteristic and integral inequalities with maximal radial characteristic for meromorphic functions and for differences of subharmonic functions

Abstract

Let f f be a meromorphic function on the complex plane with Nevanlinna characteristic T ( r , f ) T(r,f) and maximal radial characteristic ln ⁡ M ( t , f ) \ln M(t,f) , where M ( t , f ) M(t,f) is the maximum of the modulus | f | |f| on circles centered at zero and of radius t t . A number of well-known and widely used results make it possible to estimate from above the integrals of ln ⁡ M ( t , f ) \ln M (t,f) over subsets E E on segments [ 0 , r ] [0,r] in terms of T ( r , f ) T(r,f) and the linear Lebesgue measure of E E . In the paper, such estimates are obtained for the Lebesgue–Stieltjes integrals of ln ⁡ M ( t , f ) \ln M(t,f) with respect to an increasing integration function m m , and the sets E E on which the function m m is not constant can have fractal nature. At the same time, it is possible to obtain nontrivial estimates in terms of the h h -content and h h -Hausdorff measure of the set E E , as well as their partial d d -dimensional power versions with d ∈ ( 0 , 1 ] d\in (0,1] . All preceding similar estimates known to the author correspond to the extreme case of d = 1 d=1 and an absolutely continuous integration function m m with density of class L p L^p for p > 1 p>1 . The main part of the presentation is carried out immediately for the differences of subharmonic functions, or δ \delta -subharmonic functions, on circles centered at zero with explicit constants in the estimates. The only restriction in the main theorem is that the modulus of continuity of the function m m should satisfy the Dini condition at zero, and this condition, as a counterexample shows, is essential.