On Reverse Pinsker Inequalities

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.