The holomorphic solutions of the generalized Dhombres functional equation

We study holomorphic solutions f of the generalized Dhombres equation f(zf(z))=φ(f(z)), z∈C, where φ is in the class E of entire functions. We show, that there is a nowhere dense set E0⊂E such that for every φ∈E∖E0, any solution f vanishes at 0 and hence, satisfies the conditions for local analytic solutions with fixed point 0 from our recent paper. Consequently, we are able to provide a characterization of solutions in the typical case where φ∈E∖E0. We also show that for polynomial φ any holomorphic solution on C∖{0} can be extended to the whole of C. Using this, in special cases like φ(z)=zk+1, k∈N, we can provide a characterization of the analytic solutions in C.