Let　B-n(Omega)　be　the　set　of　Cowen-Douglas　operators　of　index　n　on　a　nonempty　bounded　connected　open　subset　Omega　of　C.　We　consider　the　strong　irreducibility　of　a　class　of　Cowen-Douglas　operators　F　B-n(Omega)　on　Banach　spaces.　We　show　F　B-n(Omega)　subset　of　B-n(Omega)　and　give　some　conditions　under　which　an　operator　T　is　an　element　of　F　B-n(Omega)　is　strongly　irreducible.　All　these　results　generalise　similar　results　on　Hilbert　spaces.