# zbMATH — the first resource for mathematics

New proof of the Semmes inequality for the derivative of the rational function. (English. Russian original) Zbl 1130.30312
Math. Notes 72, No. 2, 230-236 (2002); translation from Mat. Zametki 72, No. 2, 258-264 (2002).
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.

##### MSC:
 30D45 Normal functions of one complex variable, normal families 30D55 $$H^p$$-classes (MSC2000)
Full Text: