Meromorphic functions and their derivatives: equivalence of norms Konstantin Dyakonov 30D4530D5030D55inner functionsstar-invariant subspacesdifferentiationreverse Bernstein inequalityToeplitz operators For an inner function $\theta$ on the upper half-plane $\mathbb{C}_{+}$, we look at the star-invariant subspace $K^{p}_{\theta} := H^{p} \cap \theta \overline{H^{p}}$ of the Hardy space $H^{p}$. We characterize those $\theta$ for which the differentiation operator $f \mapsto f'$ provides an isomorphism between $K^{p}_{\theta}$ and a closed subspace of $H^{p}$, with $1 < p < \infty$. Namely, we show that such $\theta$'s are precisely the Blaschke products whose zero-set lies in some horizontal strip $\{a < \mathfrak{I}z < b \}$, with $0 < a < b < \infty$, and splits into finitely many separated sequences. We also describe the case of a single separated sequence in terms of the left inverse to the differentiation map; the description involves coanalytic Toeplitz operators. While our main result provides a criterion for the $H^{p}$-norms $\| f \|_{p}$ and $\| f' \|_{p}$ to be equivalent (written as $\| f \|_{p} \asymp \| f' \|_{p}$), where $f$ ranges over a certain family of meromorphic functions with fixed poles, some other spaces $Y$ that admit a similar estimate $\| f \|_{Y} \asymp \| f' \|_{Y}$ under similar conditions are also pointed out. Indiana University Mathematics Journal 2008 text pdf 10.1512/iumj.2008.57.3320 10.1512/iumj.2008.57.3320 en Indiana Univ. Math. J. 57 (2008) 1557 - 1572 state-of-the-art mathematics http://iumj.org/access/