Local analytic solutions of a more generalized dhombres equation

We study the local analytic solutions f of the functional equation f(ψ(zf(z))) = ϕ (f(z)) for z in some neighborhood of the origin. Whether the solution f vanishes at z=0 or not plays a critical role for local analytic solutions of this equation. In this paper, we obtain results of analytic solutions not only in the case f(0)=0 but also for f(0)≠ 0. When assuming f(0)=0, for technical reasons, we just get the result for f′(0)≠0. Then when assuming f(0)=ω0≠0, ψ′(0)=s≠0, ψ(z) is analytic at z=0 and ϕ(z) is analytic at z=ω0, we give the existence of local analytic solutions f in the case of 0<|sω_0|<1 and the case of |sω0|=1 with the Brjuno condition.