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The initial motivation for this paper is to discuss a more concrete approachto an approximation theorem of Axler and Shields, which says that the uniformalgebra on the closed unit disc closure(D) generated by z and h --- where h isa nowhere-holomorphic harmonic function on D that is continuous up to theboundary --- equals the algebra of continuous functions on closure(D). Theabstract tools used by Axler and Shields make harmonicity of h an essentialcondition for their result. We use the concepts of plurisubharmonicity andpolynomial convexity to show that, in fact, the same conclusion is reached if his replaced by h+R, where R is a non-harmonic perturbation whose Laplacian is"small" in a certain sense.

Uniform algebras generated by holomorphic and close-to-harmonic functions