Analytic centers and analytic diameters of planar continua

Abstract:

This paper contains some basic results about analytic centers and analytic diameters, concepts which arise in Vitushkin’s work on rational approximation. We use Carathéodory’s theorem to calculate $\beta (K,z)$ in the case in which K is a continuum in the complex plane. This leads to the result that, if $g:\Omega (K) \to \Delta (0;1)$ is the normalized Riemann map, then $\beta (g,0)/\gamma (K)$ is the unique analytic center of K and $\beta (K) = \gamma (K)$. We also give two proofs of the fact that $\beta (g,0)/\gamma (K) \in {\text {co}}\;(K)$. We use Bieberbach’s and Pick’s theorems to obtain more information about the geometric location of the analytic center. Finally, we obtain inequalities for $\beta (E,z)$ for arbitrary bounded planar sets E.