On the zero-free polynomial approximation problem

Let EE be a compact set in CC with connected complement, and let A(E)A(E) be the class of all complex continuous function on EE that are analytic in the interior E0E0 of EE. Let f∈A(E)f∈A(E) be zero free on E0E0. By Mergelyan’s theorem ff can be uniformly approximated on EE by polynomials, but is it possible to realize such approximation by polynomials that are zero-free on EE? This natural question has been proposed by J. Andersson and P. Gauthier. So far it has been settled for some particular sets EE. The present paper describes classes of functions for which zero free approximation is possible on an arbitrary EE.