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Cauchy transforms of point masses: the logarithmic derivative of polynomials. (English) Zbl 1099.30023
The authors obtain sharp estimates for the 1-dimensional Hausdorff measure of the sets \[ \left\{z:~\left| \sum_{k=1}^N\frac1{z-z_k}\right| >P\right\} \] and \[ \left\{z:~\sum_{k=1}^N\frac1{| z-z_k| }>P\right\}, \] where the \(z_k\) are points in the complex plane and \(P>0\).

MSC:
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
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