Famous public school rejected by Duke (5)
I believe the answer is:
noted
'famous' is the definition.
(noted can mean well-known or famous)
'public school rejected by duke' is the wordplay.
'public school' becomes 'eton' (Eton College).
'rejected' says the letters should be written backwards.
'by' means one lot of letters go next to another.
'duke' becomes 'd' (abbreviation).
'eton' backwards is 'note'.
'note'+'d'='NOTED'
(Other definitions for noted that I've seen before include "Remarked - famous" , "Famed" , "taken down" , "Distinguished" , "Observed - famous" .)