Let $f(z) = p(z) / q(z)$ be a rational function of degree at most $m$ (i.e. $\deg p \leq m$ and $\deg q \leq m$). Suppose $f$ is holomorphic in the closed unit disk and therefore has an expansion $f(z) = \sum_{n \geq 0} a_n z^n$ valid for $|z| \leq 1$. It is not hard to see that $f$ has finite Hardy norm ($\|f\|_{H^2}^2 = \sum_{n \geq 0} |a_n|^2$) and finite Dirichlet norm ($\|f\|_{\mathcal{D}}^2 = \sum_{n \geq 0} (n+1) |a_n|^2$).
Does anyone know an upper bound for the maximal distortion between these two norms $$ \sup_f \frac{\|f\|_{\mathcal{D}}}{\|f\|_{H^2}} , $$ when the supremum is taken over all rational functions of degree at most $m$ holomorphic in the unit disk?
I would also be interested in similar bounds requiring further assumptions on the class of functions (e.g. rational functions with poles far away from the unit disk).