2014/09/17 by Włodzimierz Zwonek, Zbigniew Błocki, Zwonek, Włodzimierz +1
Mathematics · #32A25 #32F45 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.1409.5023
openalex publication_date 2014/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It has been recently shown that for a convex domain Ω in \mathbb Cn and w∈Ω the function FΩ(w):=(KΩ(w)λ(IΩ(w)))1/n, where KΩ is the Bergman kernel on the diagonal and IΩ(w) the Kobayashi indicatrix, satisfies 1≤ FΩ≤ 4. While the lower bound is optimal, not much more is known about the upper bound. In general it is quite difficult to compute FΩ even numerically and the highest value of it obtained so far is 1.010182… In this paper 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 had been known, the main point is to compute the volume of the Kobayashi indicatrix. It turns out that in the second case the function λ(IΩ(w)) is not C3,1.