New proof of the Semmes inequality for the derivative of the rational function (Q1810182)

For a function \(f\) analytic on the open unit disc \(D\) and \(\alpha> 0\), by \({\mathcal T}^\alpha f\) we denote the Riemann-Liouville derivative of order \(\alpha\). \(H_p\) \((0< p\leq \infty)\) denotes the Hardy space and \(B_p\) \((0< p\leq\infty)\) denotes the Besov space. Let \(H^\alpha_p\) be the set of \(f\) with \({\mathcal T}^\alpha f\in H_p\) and \(\| f\|_{H^\alpha_p}= \|{\mathcal T}^\alpha f\|_{H_p}\), and \(B^\alpha_p\) the set of \(f\) with \({\mathcal T}^\alpha f\in B^\alpha_p\) and \(\| f\|_{B^\alpha_p}= \|{\mathcal T}^\alpha f\|_{B_p}\). The author shows that if all the poles of the rational function \(R\) of degree \(n\), \(n= 1,2,3,\dots\), lie in \(\mathbb C\setminus D\), then \(\| R\|_{H^\alpha_{1/2}}\leq cn^\alpha\| R\|_\beta\), \(\| R\|_{\beta^\alpha_{1/2}}\leq cn^\alpha\| R\|_\beta\), where \(\alpha> 0\) and \(c> 0\) depends only on \(\alpha\). For the proof, the author uses the special integral representation of rational functions.