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 Verdú for general probability measures. A second bound improves the tightness of an inequality by Csiszár 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 Rényi divergence of an arbitrary non-negative order (including $\infty$) as a function of the total variation distance. Another lower bound by Verdú 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.