On generalized Dhombres equations with non-constant rational solutions in the complex plane

We characterize the generalized Dhombres functional equations f ( z f ( z ) ) = φ ( f ( z ) ) which have a non-constant rational solution f 0 , holomorphic at 0, with f 0 ( 0 ) = w 0 , under the assumption that φ is a non-constant meromorphic function, holomorphic at w 0 . All rational solutions of such an equation which are holomorphic at 0 and all formal solutions are determined.