  We establish a reversal of Lyapunov's inequality for monotone log-concave
sequences, settling a conjecture of Havrilla-Tkocz and Melbourne-Tkocz. A
strengthened version of the same conjecture is disproved through counter
example. We also derive several information theoretic inequalities as
consequences. In particular sharp bounds are derived for the varentropy,
R\'enyi entropies, and the concentration of information of monotone log-concave
random variables. Moreover, the majorization approach utilized in the proof of
the main theorem, is applied to derive analogous information theoretic results
in the symmetric setting, where the Lyapunov reversal is known to fail.
