New Proof of the Semmes Inequality for the Derivative of the Rational Function

In the open disk $|z|<1$ of the complex plane, we consider the following spaces of functions: the Bloch space $\mathscr B$; the Hardy–Sobolev space $H^\alpha _p$, $\alpha \ge 0$, $0<p\le \infty$; and the Hardy–Besov space $B^\alpha _p$, $\alpha \ge 0$, $0<p\le \infty$. It is shown that if all the poles of the rational function $R$ of degree $n$, $n=1,2,3,\dots$, lie in the domain $|z|>1$, then $\|R\|_{H^\alpha _{1/\alpha }}\le cn^\alpha \|R\|_{\mathscr B}$, $\|R\|_{B^\alpha _{1/\alpha }}\le cn^\alpha \|R\|_{\mathscr B}$, where $\alpha >0$ and $c >0$ depends only on $\alpha$ . The second of these inequalities for the case of the half-plane was obtained by Semmes in 1984. The proof given by Semmes was based on the use of Hankel operators, while our proof uses the special integral representation of rational functions.