Asymptotic sharpness of a Bernstein-type inequality for rational functions in $H^2$

A Bernstein-type inequality for the standard Hardy space $H^2$ in the unit disk $\mathbb D=\{z\in\mathbb C\colon|z|<1\}$ is considered for rational functions in $\mathbb D$ having at most $n$ poles all outside of $\frac1r\mathbb D$, $0<r<1$. The asymptotic sharpness is shown as $n\to\infty$ and $r\to1$.