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Capacity Estimates for Planar Cantor-Like Sets

Published online by Cambridge University Press:  20 November 2018

Carl David Minda
Carl David Minda*
Affiliation:
University of Cincinnati, Cincinnati, Ohio
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Upper and lower bounds for the capacity of planar Cantor-like sets are presented. Chebichev polynomials are the principal tool employed in the derivation of these estimates. A necessary and sufficient condition for certain planar Cantor-like sets to have positive capacity is obtained. Related one-sided capacitary estimates for more general Cantor-like sets can be found in [3, pp. 106-109]. Techniques analogous to those used in this paper yield similar results for linear Cantor-like sets which are well-known [2, pp. 150-161]. The use of Chebichev polynomials to obtain these results provides an alternate, possibly more elementary, approach to these linear problems.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 5 , October 1974 , pp. 1169 - 1172
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Goluzin, G. M., Geometric theory of functions of a complex variable. Translations of Mathematical Monographs, vol. 26 (Amer. Math. Soc, Providence, 1969).Google Scholar
2. Nevanlinna, R., Analytic functions, Die Grundlehren der math. Wissenschaften, Band 162 (Springer-Verlag, New York, 1970).Google Scholar
3. Tsuji, M., Potential theory in modern function theory (Maruzen, Tokyo, 1959).Google Scholar