Well-known public school rejected by duke (5)
I believe the answer is:
noted
'well-known' is the definition.
(thesaurus)
'public school rejected by duke' is the wordplay.
'public school' becomes 'eton' (I've seen this before).
'rejected' says the letters should be written in reverse.
'by' says to put letters next to each other.
'duke' becomes 'd'.
'eton' reversed gives 'note'.
'note'+'d'='NOTED'
(Other definitions for noted that I've seen before include "Gaiter; oyster spawn" , "Made a record - celebrated" , "Famous or eminent" , "Celebrity" , "Observed or took account of" .)