Integrate inequalities for the higher derivatives of Blashke product

T. S. Mardvilko

Journal of the Belarusian State University. Mathematics and Informatics, Tome 1 (2018), pp. 10-16

Résumé

Upper and lower inequalities for the higher derivatives of Blashke product in the Lebesgue space $L_{p}$ are obtained in this work. All $p\in (0,+\infty)\setminus \{\frac{1}{s}\}, s\in \mathbb{N}\setminus \{1\}$, are considered, where s is order of the derivative. The case $p = \frac{1}{s}$ was investigated by the author earlier. Let $a_{n}=\{a_{1},\dots , a_{n}\}$ be a certain set of $n$ complex numbers laying in the unit disc $|z| 1$. Let us introduce the Blashke products $b_{n}(z)=\displaystyle\prod_{k=1}^{n} \frac{z-a_{k}}{1-\bar{a_{k}}z}$ with zeros at the points $a_{1}, a_{2},\dots , a_{n}$. For $0$ and $s\in \mathbb{N}$ holds the equality $\displaystyle\inf_{a_{n}}\lVert b_{n}^{(s)}\rVert_{L_{p}}=0$. For $p>1$ $\displaystyle\inf_{a_{n}}\lVert b_{n}^{'}\rVert_{L_{p}}=n$. For $\frac{1}{s}$ and $s\in \mathbb{N}$ holds the equality $\displaystyle\sup_{a_{n}}\lVert b_{n}^{(s)}\rVert_{L_{p}}=+\infty$. In other cases, the obtained estimates are exact in order. The main results of the present paper are stated in theorems $1 - 5$.

Détail

Keywords: Blashke product; rational functions; higher derivatives; Lebesgue space.