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.
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