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. © 2006 Elsevier Inc. All rights reserved.