An upper bound for the logarithmic capacity of two intervals

TL;DR

This paper derives an elementary upper bound for the logarithmic capacity of two real intervals using inequalities for elliptic and theta functions, improving understanding of these capacities.

Contribution

It provides a new elementary upper bound for the logarithmic capacity of two intervals, building on classical elliptic function representations.

Findings

Derived an explicit upper bound in elementary functions

Improved bounds for specific interval configurations

Connected elliptic function inequalities to capacity estimates

Abstract

The logarithmic capacity (also called Chebyshev constant or transfinite diameter) of two real intervals $[-1,\alpha]\cup[\beta,1]$ has been given explicitly with the help of Jacobi's elliptic and theta functions already by Achieser in 1930. By proving several inequalities for these elliptic and theta functions, an upper bound for the logarithmic capacity in terms of elementary functions of $\alpha$ and $\beta$ is derived.