Abstract:
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 $0<p<\frac{1}{s}$ 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 $\frac{1}{s}<p<\infty$ 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 $1 - 5$.
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