On the zero-free polynomial approximation problem

Let $E$ be a compact set in $\mathbb C$ with connected complement, and let $A(E)$ be the class of all complex continuous function on $E$ that are analytic in the interior $E^0$ of $E$. Let $f \in A(E)$ be zero free on $E^0$. By Mergelyan's theorem $f$ can be uniformly approximated on $E$ by polynomials, but is it possible to realize such approximation by polynomials that are zero-free on $E$? This natural question has been proposed by J. Andersson and P. Gauthier. So far it has been settled for some particular sets $E$. The present paper describes classes of functions for which zero free approximation is possible on an arbitrary $E$.