On the first two coefficients of the Bergman function expansion for radial metrics

Let g(F) be Miller metrics on rotation invariant domains Omega = B-d, C-d, B-d* C-d* associated with the Mier potentials Phi(F)(z, (z) over barY) = F(In(parallel to z parallel to(2)). The purpose of this paper is two fold. Firstly, we obtain explicit formulas of the coefficients a(j) (j = 1, 2) of the Bergman function expansion for the domains (Omega, g(F)). Secondly, we obtain explicit expressions of F when both a(1) and a(2) are constants on Omega. (C) 2017 Elsevier B.V. All rights reserved.