Two-sided bounds for the logarithmic capacity of multiple intervals

TL;DR

This paper provides simple, precise bounds for the logarithmic capacity of multiple intervals on the real line, improving previous estimates using advanced potential theory techniques.

Contribution

It introduces new upper and lower bounds for the logarithmic capacity of multiple intervals, including cases with infinitely many intervals, using separating transformation and dissymmetrization methods.

Findings

Derived bounds are more accurate than previous estimates.

Graphical comparison shows the effectiveness of the bounds.

Results extend to sets with infinitely many intervals.

Abstract

Potential theory on the complement of a subset of the real axis attracts a lot of attention both in function theory and applied sciences. The paper discusses one aspect of the theory - the logarithmic capacity of closed subsets of the real line. We give simple but precise upper and lower bounds for the logarithmic capacity of multiple intervals and a lower bound valid also for closed sets comprising an infinite number of intervals. Using some known methods to compute the exact values of capacity we demonstrate graphically how our estimates compare with them. The main machinery behind our results are separating transformation and dissymmetrization developed by V.N. Dubinin and a version of the latter by K. Haliste as well as some classical symmetrization and projection result for logarithmic capacity. The results of the paper improve some previous achievements by A.Yu. Solynin and K. Shiefermayr.