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The family of Cauchy transforms \[C_{g}(z,w) = -\frac{1}{\pi}\int_{\mathbb{C} } \frac{g(u)}{\overline{u-w} (u-z) } da(u ),\] where the measurable function $g$ with compact (essential) support satisfies $0 \leq g\leq 1,$ and suitably defined for all complex $z, w,$ is closely connected to the theory of Hilbert space operators with one-dimensional self-commutators. Based on these connections one can derive the inequality \[\vert 1-\exp C{g}(z,w)\vert\leq 1. \] Here, using elementary methods, a direct proof of this inequality is given. The approach involves a detailed study of the convex family of integrals \[I_{g}= -\frac{1}{\pi}\int_{\mathbb{C} } \frac{g(u)}{\overline{u+1} (u-1) } da(u),\] where $g$ varies over the set of measurable functions with compact support satisfying $0 \leq g\leq 1.$ These integrals are transformed to a tractable form using a parametriztion of the plane minus the real axis using the family of circles passing though the points $+1,-1.$ The characeristic functions of discs bounded by these circles are unique points in the boundary of the convex set of values of the family of integrals.

Author

Clancey, Kevin F.

Published

2022

Language

EN