The values of a family of Cauchy transforms

Abstract: 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.