  New upper bounds on the relative entropy are derived as a function of the
total variation distance. One bound refines an inequality by Verd\'{u} for
general probability measures. A second bound improves the tightness of an
inequality by Csisz\'{a}r and Talata for arbitrary probability measures that
are defined on a common finite set. The latter result is further extended, for
probability measures on a finite set, leading to an upper bound on the
R\'{e}nyi divergence of an arbitrary non-negative order (including $\infty$) as
a function of the total variation distance. Another lower bound by Verd\'{u} on
the total variation distance, expressed in terms of the distribution of the
relative information, is tightened and it is attained under some conditions.
The effect of these improvements is exemplified.
