Existence of analytic solutions of an iterative functional equation

This paper is concern analytic solutions of an iterative functional equation of the formf(p(z)+q(f(z)))=h(f(z)),z∈C.f(p(z)+q(f(z)))=h(f(z)),z∈C.Byconstructing a convergent power series solution of an auxiliary equationg(ϕ(α2z)-p(ϕ(αz)))=h(g(ϕ(αz)-p(ϕ(z)))),z∈C,g(ϕ(α2z)-p(ϕ(αz)))=h(g(ϕ(αz)-p(ϕ(z)))),z∈C,analytic solutions of the original equation are obtained. We discuss not only these α appeared in the auxiliary equation at the hyperbolic case 0 < ∣α∣ ≠ 1 and resonance, i.e., at a root of the unity, but also those α near resonance (i.e., near a root of the unity) under Brjuno condition.