Analytic centers and analytic diameters of planar continua

Source 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 β ( K , z ) \beta (K,z) in the case in which K is a continuum in the complex plane. This leads to the result that, if g : Ω ( K ) → Δ ( 0 ; 1 ) g:\Omega (K) \to \Delta (0;1) is the normalized Riemann map, then β ( g , 0 ) / γ ( K ) \beta (g,0)/\gamma (K) is the unique analytic center of K and β ( K ) = γ ( K ) \beta (K) = \gamma (K) . We also give two proofs of the fact that β ( g , 0 ) / γ ( K ) ∈ co ( K ) \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 β ( E , z ) \beta (E,z) for arbitrary bounded planar sets E .