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We establish a reversal of Lyapunov's inequality for monotone log-concavesequences, settling a conjecture of Havrilla-Tkocz and Melbourne-Tkocz. Astrengthened version of the same conjecture is disproved through counterexample. We also derive several information theoretic inequalities asconsequences. In particular sharp bounds are derived for the varentropy,R\'enyi entropies, and the concentration of information of monotone log-concaverandom variables. Moreover, the majorization approach utilized in the proof ofthe main theorem, is applied to derive analogous information theoretic resultsin the symmetric setting, where the Lyapunov reversal is known to fail.

A discrete complement of Lyapunov's inequality and its information theoretic consequences