A very low increase in brief (7)
I believe the answer is:
apprise
'brief' is the definition.
(thesaurus)
'a very low increase' is the wordplay.
'very low' becomes 'pp' (musical abbreviation for pianissimo, very quiet).
'increase' becomes 'rise' (rise is a kind of increase).
'a'+'pp'+'rise'='APPRISE'
'in' is the link.
(Other definitions for apprise that I've seen before include "Inform, make known to" , "Provide with information" , "Inform or tell" , "Formally inform, notify" , "tip off" .)