Relations between certain domains in the complex plane and polynomial approximation in the domains

Abstract

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 isproperly contained in the class of John disks. We give theconditions which make the converses of the above results be true.Next, we show that an exterior quasiconvex domain satisfies certain growth conditions for theexterior Riemann mapping. From the results we show that the domainsatisfies the Bernstein inequality and the integrated version ofit.Finally, we assume that f is a function which is continuous in the closure of a domainD and analytic in D. We show connections between thesmoothness of f and the rate at which it can be approximatedby polynomials on an exterior quasiconvex domain and aLip_{alpha}-extension domain.