Abstract
This paper is concern analytic solutions of an iterative functional equation of the form f ( p ( z ) + q ( f ( z ) ) ) = h ( f ( z ) ) , z ∈ C . Byconstructing a convergent power series solution of an auxiliary equation g ( ϕ ( α 2 z ) - p ( ϕ ( α z ) ) ) = h ( g ( ϕ ( α z ) - 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.
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