Abstract
It was recently shown that for a convex domain Ω in and w ∈ Ω, the function
, where
is the Bergman kernel on the diagonal and
the Kobayashi indicatrix, satisfies
. While the lower bound is optimal, not much more is known about the upper bound. In general, it is quite difficult to compute
even numerically, and the largest value of it obtained so far is 1.010182… . In this article, we present precise, although rather complicated, formulas for the ellipsoids Ω = {|z1|2m + |z2|2 < 1} (with m ≥ 1/2) and all w, as well as for Ω = {|z1| + |z2| < 1} and w on the diagonal. The Bergman kernel for those ellipsoids was already known; the main point is to compute the volume of the Kobayashi indicatrix. It turns out that in the second case, the function
is not C3, 1.