  We extend and simplify results of \cite{Din~2009} where the asymptotic
behavior of the holomorphic sectional curvature of the Bergman metric in annuli
is studied. Similarly as in \cite{Din~2009} the description enables us to
construct an infinitely connected planar domain (in our paper it is a Zalcman
type domain) for which the supremum of the holomorphic sectional curvature is
two whereas its infimum is equal to $-\infty$.
