Let G be a bounded simply
connected domain in the complex plane. Using a result of Hedberg, we show that the
polynomials are dense in Bergman space La2(G) if G is the image of the unit disk U
under a weak-star generator of H∞. We also show that density of the polynomials in
La2(G) implies density of the polynomials in H2(G). As a consequence,
we obtain new examples of cyclic analytic Toeplitz operators on H2(U)
and composition operators with dense range on H2(U). As an additional
consequence, we show that if the polynomials are dense in La2(G) and ϕ maps U
univalently onto G, then ϕ is univalent almost everywhere on the unit circle
C.
