Let P denote the Bergman projection on the unit disk D, P mu(z) := integral(D) mu(w)/(1-z (w) over bar)(2) dA(w), z is an element of D, where dA is normalized area measure. We prove that if vertical bar mu(z)vertical bar <= 1 on D, then the integral I(mu()a,r) := integral(2 pi)(0) exp {a r(4)vertical bar P mu(re(i theta))vertical bar(2)/log 1/1-r(2)}d theta/2 pi, 0 < r < 1, has the bound I-mu(a,r) <= C(a) := 10(1-a)(-3/2) for 0 < a < 1, irrespective of the choice of the function mu. Moreover, for a > 1, no such uniform bound is possible. We interpret the theorem in terms the asymptotic tail variance of such a Bergman projection P-mu (by the way, the asymptotic tail variance induces a seminorm on the Bloch space). This improves upon earlier work of Makarov, which covers the range 0 < a < pi(2)/64 = 0.1542.... We then apply the theorem to obtain an estimate of the universal integral means spectrum for conformal mappings with a k-quasiconformal extension, for 0 < k < 1. The estimate reads, for t is an element of C and 0 < k < 1, B(k,t) <= {1/4 k(2)vertical bar t vertical bar(2)(1 + 7k)(2), for vertical bar t vertical bar <= 2/k(1 + 7k)(2), k vertical bar t vertical bar - 1/(1 + 7k)(2), for vertical bar t vertical bar >= 2/k(1 + 7k)(2), which should be compared with the conjecture by Prause and Smirnov to the effect that for real t with vertical bar t vertical bar <= 2/k, we should have B(k, t) = 1/4k(2)t(2).
QC 20170619