  The line of investigation of the present paper goes back to a classical work
of W. H. Gustafson of the 1973, in which it is described the probability that
two randomly chosen group elements commute. In the same work, he gave some
bounds for this kind of probability, providing information on the group
structure. We have recently obtained some generalizations of his results for
finite groups. Here we improve them in the context of the compact groups.
