A convolution approach on partial sums of certain analytic and univalent functions.
Agler, Lykova and Young introduced a sequence , where ν ≥ 0, of necessary conditions for the solvability of the finite interpolation problem for analytic functions from the open unit disc into the symmetrized bidisc Γ. They conjectured that condition is necessary and sufficient for the solvability of an n-point interpolation problem. The aim of this article is to give a counterexample to that conjecture.
Applying results of the infinitary Ramsey theory, namely the dichotomy principle of Galvin-Prikry, we show that for every sequence of scalars, there exists a subsequence such that either every subsequence of defines a universal series, or no subsequence of defines a universal series. In particular examples we decide which of the two cases holds.
We give a distortion theorem for quasiconformal automorphisms of the unit disk and its application to improving some results due to Douady and Earle.