The values of a family of Cauchy transforms

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 $0\le g\le 1,$ 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 $$\mathrm{\mid }1-\exp Cg(z,w)\mathrm{\mid }\le 1.$$ 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 $0\le g\le 1.$ 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.