Bloch functions and asymptotic tail variance

We obtain an optimal exponential square integrability theorem for the Bergman projection of a function bounded by 1 in modulus. This is interpreted as the statement that the asymptotic tail variance of such a function is at most 1. The asymptotic tail variance defines a seminorm on the Bloch space. We apply the main result to quasiconformal Teichm"uller theory, and obtain an estimate of the integral means spectrum of k-quasiconformal mappings that are conformal in the exterior disk: $B(k,t)\le \frac{1}{4}{k}^2\mathrm{\mid }t{\mathrm{\mid }}^2(1+7k{)}^2$. This is conjectured asymptotically sharp as k tends to 0, by Prause and Smirnov (2011).