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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Analytic centers and analytic diameters of planar continua


Author: Steven Minsker
Journal: Trans. Amer. Math. Soc. 191 (1974), 83-93
MSC: Primary 30A82
MathSciNet review: 0361094
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0361094-4
PII: S 0002-9947(1974)0361094-4
Article copyright: © Copyright 1974 American Mathematical Society