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' is a reversal indicator.
'by' means one lot of letters go next to another.
'duke' becomes 'd' (abbreviation).
'eton' written backwards gives 'note'.
'note'+'d'='NOTED'
(Other definitions for noted that I've seen before include "Remarked - famous" , "Famed" , "taken down" , "Distinguished" , "Observed - famous" .)