  The Schwarz lemmas are well-known characterizations for holomorphic maps and
we exhibit two examples of their applications. For a sequence family of
biholomorphisms $f_j$, it is useful to determine the location of $f_j(q)$ for a
fixed point $q$ in source manifolds (see Proposition \ref{2.5}). With it, we
extend the Fornaess-Stout's theorem of \citep{FS77} in monotone unions of balls
to ellipsoids in Section \ref{sec2}. In Section \ref{sec3}, we discuss the
curvature bounds of complete K\"ahler metric on $\rtimes$ domains defined in
\citep{Liu004} with an idea of \citep{Ya76}.
