  The relative log-concavity ordering $\leq_{\mathrm{lc}}$ between probability
mass functions (pmf's) on non-negative integers is studied. Given three pmf's
$f,g,h$ that satisfy $f\leq_{\mathrm{lc}}g\leq_{\mathrm{lc}}h$, we present a
pair of (reverse) triangle inequalities: if $\sum_iif_i=\sum_iig_i<\infty,$
then \[D(f|h)\geq D(f|g)+D(g|h)\] and if $\sum_iig_i=\sum_iih_i<\infty,$ then
\[D(h|f)\geq D(h|g)+D(g|f),\] where $D(\cdot|\cdot)$ denotes the
Kullback--Leibler divergence. These inequalities, interesting in themselves,
are also applied to several problems, including maximum entropy
characterizations of Poisson and binomial distributions and the best binomial
approximation in relative entropy. We also present parallel results for
continuous distributions and discuss the behavior of $\leq_{\mathrm{lc}}$ under
convolution.
