Pen an inventory for the hairdresser (7)
I believe the answer is:
stylist
'the hairdresser' is the definition.
('stylist' can be a synonym of 'hairdresser')
'pen an inventory' is the wordplay.
'pen' becomes 'sty' (both can mean an animal enclosure).
'an inventory' becomes 'list' (I've seen this in another clue).
'sty'+'list'='STYLIST'
'for' acts as a link.
(Other definitions for stylist that I've seen before include "Designer of fashionable clothing" , "Fashion expert" , "Clothes designer; hairdresser" , "Dress designer" , "(Hair) designer" .)