On the closure of the sum of two uniform algebras on compact sets in ℂ

Necessary and sufficient conditions on a compact set X in ℂ and a self-homeomorphism ψ of the plane ℂ are studied under which any function continuous on X can be approximated uniformly on X by functions of the form p + h ∘ ψ, where p is a polynomial in a complex variable and h is a rational function whose poles belong to the bounded components of the complement to the compact set ψ(X).