Håkan Hedenmalm: Bloch functions, asymptotic variance, and zero packing

We study the asymptotic variance of Bloch functions given as Bergman projections of bounded functions. This connects with quasiconformal theory and work of Ivrii shows that it gives the main coefficient for the integral means spectrum of quasiconformal mappings with small Beltrami coefficient. We show that the universal asymptotic variance is < 1, which shows that the conjecture of Prause and Smirnov is incorrect (in the better direction) and hence the conjecture of Astala that \(D(k)=1+k^2\) fails as well.