Abstract view
Remark on integral means of derivatives of Blaschke products


Published:20171005
Printed: Sep 2018
Atte Reijonen,
Department of Physics and Mathematics, University of Eastern Finland, P.O.Box 111, FI80101 Joensuu, Finland
Abstract
If $B$ is the Blachke product with zeros $\{z_n\}$, then $B'(z)\le
\Psi_B(z)$, where
$$\Psi_B(z)=\sum_n \frac{1z_n^2}{1\overline{z}_nz^2}.$$
Moreover, it is a wellknown fact that, for $0\lt p\lt \infty$,
$$M_p(r,B')=
\left(\frac{1}{2\pi}\int_{0}^{2\pi} B'(re^{i\t})^p\,d\t
\right)^{1/p}, \quad 0\le r\lt 1,$$
is bounded if and only if $M_p(r,\Psi_B)$ is bounded.
We find a Blaschke product $B_0$ such that $M_p(r,B_0')$ and
$M_p(r,\Psi_{B_0})$ are not comparable for any $\frac12\lt p\lt \infty$.
In addition, it is shown that, if $0\lt p\lt \infty$, $B$ is a CarlesonNewman
Blaschke product and a weight $\omega$ satisfies a certain regularity
condition, then
$$
\int_\mathbb{D} B'(z)^p\omega(z)\,dA(z)\asymp \int_\mathbb{D} \Psi_B(z)^p\omega(z)\,dA(z),
$$
where $dA(z)$ is the Lebesgue area measure on the unit disc.