Function algebras on disks II

This work is a continuation of [7]. In that paper, a sufficient condition was given on a real analytic fmlction g defined near 0 in C so that the algebra generated by z² and g² is dense in the space of continuous functions on D for all disks D close enough to the origin in C. By using the same methods and some ideas taken from the first named author's thesis we deal with the case where g is only of class C¹ near 0.