REMARKS ON SOME CLASSES OF POSITIVE CONTINUOUS FUNCTIONS IN C^n

Abstract

Here we prove two propositions providing sufficient conditions of belonging positive continuous functions in  to classes  and  These auxiliary classes plays important role in theory of entire functions of bounded L-index in direction and bounded L-index in joint variables, where   are continuous functions. They help to constuct general theory of bounded index for very wide class of entire functions, because for every entire functions with bounded multiplicities of zero points there exists a corresponding function  or  providing boundedness of L-index in direction or boundedness  L–index in joint variables respectively. Our result requires uniform boundedness of logarithmic derivative in all variables  and  for belonging the function to class Q^n.  Another result requires uniform boundedness of logarithmic derivative in directions  and  for belonging the function to class Q^n_b  where  is the complex conjugate vector to  b.