The holomorphic solutions of the generalized Dhombres functional equation

We study holomorphic solutions f of the generalized Dhombres equation f (zf (z)) = 0(f (Z)), Z E C, where phi is in the class epsilon of entire functions. We show, that there is a nowhere dense set epsilon(0) subset of epsilon such that for every phi is an element of epsilon \ epsilon(0), 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 phi is an element of epsilon \ epsilon(0). We also show that for polynomial phi any holomorphic solution on C \ {0} can be extended to the whole of C. Using this, in special cases like phi(z) = z(k+1), k is an element of N, we can provide a characterization of the analytic solutions in C. (c) 2006 Elsevier Inc. All rights reserved.