On Chui's conjecture and approximation by simplest fractions

In 1971 C. K. Chui conjectured that the average field strength in the unit disk $\mathbb D=\{|z|<1\}$ in the complex plane due to unit point masses on the unit circle $\mathbb T=\{|z|=1\}$ is minimal for the uniform distribution of masses. Formally the Chui's conjecture says that for all $\{z_1,\ldots,z_N\}\subset\mathbb T$, $N=1,2,\ldots$, the following is satisfied
$$ \bigg\|\sum_{k=1}^{N}\frac1{z-z_k}\bigg\|_{L^1(\mathbb D)}\geqslant \bigg\|\sum_{k=1}^{N}\frac1{z-\omega_N^k}\bigg\|_{L^1(\mathbb D)}, $$
where $\omega_N$ is the principal root of unity of degree $N$, so that $\omega_N=\exp(2\pi i/N)$, and the space $L^1(\mathbb D)$ is considered with respect to the planar Lebesgue measure in $\mathbb D$.
This conjecture remains open, and in the talk we will consider its analogue for the weighted Bergman spaces $A^2_\alpha=A^2_\alpha(\mathbb D)$, $\alpha>0$. Recall, that the space $A^2_\alpha$ consists of all holomorphic function $f$ in $\mathbb D$ such that
$$ \|f\|_{2,\alpha}^2:=\frac{\alpha+1}{\pi}\int|f(z)|^2(1-|z|^2)^\alpha\,dxdy<\infty. $$

It will be shown that the statement analogous to Chui's conjecture is true for the spaces $A^2_\alpha$ for all $\alpha\in(0,1]$. In other words, for all such $\alpha$ and for all $z_1,\ldots,z_N\in\mathbb T$, $N=1,2,\ldots$, one has
$$ \bigg\|\sum_{k=1}^{N}\frac1{z-z_k}\bigg\|_{2,\alpha}\geqslant \bigg\|\sum_{k=1}^{N}\frac1{z-\omega_N^k}\bigg\|_{2,\alpha}. $$

It is planned to consider also the problem about completeness in the space $A^2_\alpha$, $\alpha>0$, of the system of ‘simplest fractions’, that is functions of the form
$$ \sum_{k=1}^{N}\frac1{z-z_k}, $$
where $z_1,\ldots,z_N\in\mathbb T$, $N=1,2,\ldots$.
This is a joint work with E. Abakumov \textup(University Paris Est, Marne-la-Vallée, France\textup) and A. Borichev \textup(Aix–Marseille University, France\textup).