  The $\alpha$-divergences include the well-known Kullback-Leibler divergence,
Hellinger distance and $\chi^2$-divergence. In this paper, we derive
differential and integral relations between the $\alpha$-divergences that are
generalizations of the relation between the Kullback-Leibler divergence and the
$\chi^2$-divergence. We also show tight lower bounds for the
$\alpha$-divergences under given means and variances. In particular, we show a
necessary and sufficient condition such that the binary divergences, which are
divergences between probability measures on the same $2$-point set, always
attain lower bounds. Kullback-Leibler divergence, Hellinger distance, and
$\chi^2$-divergence satisfy this condition.
