The Cauchy transform on bounded domains

Suppose that $f$ is in ${L^2}(\Delta )$ where $\Delta$ is the unit disk, and that $f = 0$ outside $\Delta$. We show that then the Cauchy transform $\mathcal{C}\,f$ of $f$, when restricted to $\Delta$, satisfies $\vert\vert\mathcal{C}\,f\vert{\vert _2} \leq (2/\alpha )\vert\vert f\vert{\vert _2}$, where $\alpha \approx 2.4048$ is the smallest positive zero of the Bessel function ${J_0}$. This inequality is sharp.