RELATIONS BETWEEN CERTAIN DOMAINS IN THE COMPLEX PLANE AND POLYNOMIAL APPROXIMATION IN THE DOMAINS

We show that the class of inner chordarc domains is properly contained in the class of exterior quasiconvex domains. We also show that the class of exterior quasiconvex domains is properly contained in the class of John disks. We give the conditions which make the converses of the above results be true. Next , we show that an exterior quasiconvex domain satisfies certain growth conditions for the exterior Riemann mapping. From the results we show that the domain satisfies the Bernstein inequality and the integrated version of it. Finally, we assume that f is a function which is continuous in the closure of a domain D and analytic in D. We show connections between the smoothness of f and the rate at which it can be approximated by polynomials on an exterior quasiconvex domain and a $Lip_\alpha$-extension domain.