  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.
