Application of a Bernstein-type inequality to rational interpolation in the Dirichlet space

We prove a Bernstein-type inequality involving the Bergman and Hardy norms, for rational functions in the unit disk $\mathbb D$ having at most $n$ poles all outside of $\frac1r\mathbb D$, $0<r<1$. The asymptotic sharpness of this inequality is shown as $n\to\infty$ and $r\to1^-$. We apply our Bernstein-type inequality to an efficient Nevanlinna–Pick interpolation problem in the standard Dirichlet space, constrained by the $H^2$-norm.