  In this paper, we investigate the partition inequality, joint convexity, and
Pinsker's inequality, for a divergence that generalizes the Tsallis Relative
Entropy and Kullback-Leibler divergence. The generalized divergence is defined
in terms of a deformed exponential function, which replaces the Tsallis
$q$-exponential. We also constructed a family of probability distributions
related to the generalized divergence. We found necessary and sufficient
conditions for the partition inequality to be satisfied. A sufficient condition
for the joint convexity was established. We proved that the generalized
divergence satisfies the partition inequality, and is jointly convex, if, and
only if, it coincides with the Tsallis relative entropy. As an application of
partition inequality, a criterion for the Pinsker's inequality was found.
