Relation between Hardy and Dirichlet norms for holomorphic rational functions

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).

• Perhaps I have not understood your question, but have you looked at the example $f(z)=1/(1-z)$ ?(or your function is continuous on the closed unit disk ?) – Kelenner Sep 13 '16 at 16:33
• Hi @Kelenner, I probably should have asked for $f$ to be bounded in the unit disk as well. I modified the question to make clear that the $f$'s I'm interested in will have their poles outside the closed unit disk. Thanks! – bballe Sep 14 '16 at 12:59