  This paper has two clear motivations: a technical and a practical. The
technical motivation unifies in a single and crystal clear formulation a huge
family of inequalities that have been produced separately in the last 90 years
in different contexts. But we do not just join inequalities; our method also
create a family of inequalities invisible by previous approaches. The practical
motivation is to show that our deeper approach has strength to attack various
problems. We provide new applications of our family of inequalities, continuing
the recent work by Maia et al., that, by using our main theorem, substantially
improved an inequality of Carando et al. which seemed impossible to be achieved
by their original method.
