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

A Bernstein-type inequality in the standard Hardy space H^{2} of the unit disc $\mathbb{D}$={z\in$\mathbb{C}$:,|z|<1}, for rational functions in $\mathbb{D}$ having at most n poles all outside of \frac{1}{r}$\mathbb{D}$, 0