  In the paper, necessary and sufficient conditions are presented for a
function involving a ratio of gamma functions to be logarithmically completely
monotonic. This extends and generalizes the main result in [\emph{Inequalities
and monotonicity for the ratio of gamma functions}, Taiwanese J. Math.
\textbf{7} (2003), no.~2, 239\nobreakdash--247.] and others. As applications,
several inequalities involving the volume of the unit ball in $\mathbb{R}^n$
are derived, which refine, generalize and extend some known inequalities.
