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1. CMB Online first

Reijonen, Atte
Remark on integral means of derivatives of Blaschke products
If $B$ is the Blachke product with zeros $\{z_n\}$, then $|B'(z)|\le \Psi_B(z)$, where $$\Psi_B(z)=\sum_n \frac{1-|z_n|^2}{|1-\overline{z}_nz|^2}.$$ Moreover, it is a well-known 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 Carleson-Newman 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.

Keywords:Bergman space, Blaschke product, Hardy space, integral mean
Categories:30J10, 30H10, 30H20

2. CMB Online first

Cui, Xiaohui; Wang, Chunjie; Zhu, Kehe
Area Integral Means of Analytic Functions in the Unit Disk
For an analytic function $f$ on the unit disk $\mathbb D$ we show that the $L^2$ integral mean of $f$ on $c\lt |z|\lt r$ with respect to the weighted area measure $(1-|z|^2)^\alpha\,dA(z)$ is a logarithmically convex function of $r$ on $(c,1)$, where $-3\le\alpha\le0$ and $c\in[0,1)$. Moreover, the range $[-3,0]$ for $\alpha$ is best possible. When $c=0$, our arguments here also simplify the proof for several results we obtained in earlier papers.

Keywords:logarithmic convexity, area integral mean, Bergman space, Hardy space
Categories:30H10, 30H20

3. CMB 2011 (vol 56 pp. 229)

Arvanitidis, Athanasios G.; Siskakis, Aristomenis G.
Cesàro Operators on the Hardy Spaces of the Half-Plane
In this article we study the Cesàro operator $$ \mathcal{C}(f)(z)=\frac{1}{z}\int_{0}^{z}f(\zeta)\,d\zeta, $$ and its companion operator $\mathcal{T}$ on Hardy spaces of the upper half plane. We identify $\mathcal{C}$ and $\mathcal{T}$ as resolvents for appropriate semigroups of composition operators and we find the norm and the spectrum in each case. The relation of $\mathcal{C}$ and $\mathcal{T}$ with the corresponding Ces\`{a}ro operators on Lebesgue spaces $L^p(\mathbb R)$ of the boundary line is also discussed.

Keywords:Cesàro operators, Hardy spaces, semigroups, composition operators
Categories:47B38, 30H10, 47D03

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