  We study equivalence of invariant metrics on noncompact K\"ahler manifolds
with a complete Bergman metric of bounded curvature. Especially only the
boundedness of the ratio between Bergman kernel and the $n$-times wedge product
of Bergman metric in any fundamental domain of such a K\"ahler manifold is
required to obtain the equivalence of the Bergman metric and the complete
K\"ahler--Einstein metric. To demonstrate the effectiveness of this method, we
consider a two-parameter family of $3$-dimensional bounded pseudoconvex domains
\[ E_{p,\lambda}=\{(x,y,z)\in \mathbb{C}^3 ;
(|x|^{2p}+|y|^2)^{1/{\lambda}}+|z|^2<1 \},\qquad p,\lambda>0.\] For this
family, boundary limits of the holomorphic sectional curvature of the Bergman
metric are not well-defined, and hence previously known methods for comparison
of invariant metrics do not work. Lastly, we provide an estimate of lower bound
of the integrated Carath\'eodory--Reiffen metric on complete noncompact
simply-connected K\"ahler manifolds with negative sectional curvature.
