Two-sided bounds for the logarithmic capacity of multiple intervals. (English) Zbl 1221.31004
Let \(E\) be the disjoint union of finitely many closed subintervals of \([-1,1]\). This paper gives sharp upper and lower bounds for the logarithmic capacity, \(\text{cap\,}E\), of \(E\) that improve several known estimates. For example, if
\[ E=\bigcup _{1}^{n}[a_{k},b_{k}],\quad\text{where}\quad -1=a_{1}<b_{1}<a_{2}<...<b_{n}=1\quad\text{and}\quad n\geq 2, \]
then \(\text{cap\,}E\) is at most
\[ \frac{1}{2}\left\{ \cos \left[ 2^{-1}\sum_{1}^{n-1}\left( \cos^{-1}(a_{k+1})-\cos^{-1}(b_{k})\right) \right] \right\} ^{1/(n-1)}, \] and examples are provided where equality occurs. One of the lower estimates established for \(\text{cap\,}E\) is valid even for arbitrary closed subsets \(E\) of \([-1,1]\). The bounds provided are compared graphically with the results of numerical computations.
\[ E=\bigcup _{1}^{n}[a_{k},b_{k}],\quad\text{where}\quad -1=a_{1}<b_{1}<a_{2}<...<b_{n}=1\quad\text{and}\quad n\geq 2, \]
then \(\text{cap\,}E\) is at most
\[ \frac{1}{2}\left\{ \cos \left[ 2^{-1}\sum_{1}^{n-1}\left( \cos^{-1}(a_{k+1})-\cos^{-1}(b_{k})\right) \right] \right\} ^{1/(n-1)}, \] and examples are provided where equality occurs. One of the lower estimates established for \(\text{cap\,}E\) is valid even for arbitrary closed subsets \(E\) of \([-1,1]\). The bounds provided are compared graphically with the results of numerical computations.
Reviewer: Stephen J. Gardiner (Dublin)
MSC:
| 31A15 | Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions |
Keywords:
logarithmic capacityReferences:
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