ON POSITIVE CONTINUOUS FUNCTIONS DEFINED IN THE UNIT POLYDISC

Abstract

In theory of holomorphic functions having bounded L-index in a direction b an auxiliary class of positive continuous functions L is important to describe properties of the holomorphic functions by some inequalities and
estimates containing the function L. This class is defined by local behavior of the function L. In the simplest one-dimensional case, the function
should not vary locally too quickly, i.e. $L(r+O(1/L(r))) = O(L(r))$ for $r = |z|\to+\infty$. The paper is devoted an analog of this function class for the unit polydisc, i.e. for the Cartesian product of the unit discs. There is proved an equivalence of three different approaches to define the class. It is described by the local behavior on the slice z+tb for given z from
the unit polydisc and for a fixed direction b, where the complex variable t belongs to some disc with radius dependent on b and z. These estimates must be fulfilled uniformly in all z. There is indicated a possible explicit
form of functions belonging to the class. The form is given as a product of arbitrary positive continuous function defined in the closed unit polydisc and the minimum of the expressions \(\frac{1}{1−|z_j|}\) in all variables $z_j$.