Proof of the generalized Chui conjecture

We prove Chui's conjecture on sums of Cauchy kernels. It states that among all configurations of Cauchy kernels on the unit disk, the equispaced one minimizes $L^1$ norm of the sum in the unit disk. Moreover, we obtain a generalization to a wider set of functionals, including weighted $L^p$ norms for $p > 1$ with radially symmetric weights.