The concentration index of subharmonic functions of infinite order

Abstract

The purpose of this paper is to introduce into consideration an analogue of the concentration index in the class of subharmonic functions of infinite order. The one in the case of finite order is used in the interpolation theory.We use the standard notation of the potential theory and the value distribution theory [1, 2], nevertheless we recall some of them. We denote by µ u the Riesz measure of a subharmonic function u. We put C(z, t) = {w : |w − z| ≤ t}, n(z, t) = µ u (C(z, t)), n(r) = n(0, r), and B(r, u) the maximum of u on the disk C(0, r). Without loss of generality we may assume u(0) = 0 and n(1) = 0. The set of all the subsets of [1, ∞), having finite logarithmic measure, is denoted bywhere χ S is the characteristic function of S . We denote by M positive constants.The concentration index of an entire function of finite order was introduced into consideration implicitly by Levin [3] and explicitly by Krasichkov [4], who studied its properties. The specific case of zero order was considered in [5,6].We define the concentration index I(z, u) of a subharmonic function of infinite order by the formula