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Log-concave sequences appear naturally in a variety of

fields. For example in convex geometry the Alexandrov-Fenchel

inequalities demonstrate the intrinsic volumes of a convex body to be

log-concave, while in combinatorics the resolution of the Mason

conjecture shows that the number of independent sets of size n in a

matroid form a log-concave sequence as well. By Lyapunov's

inequality we refer to the log-convexity of the (p-th power) of the

L^p norm of a function with respect to an arbitrary measure, an

immediate consequence of Holder's inequality. In the continuous

setting measure spaces satisfying concavity conditions are known to

satisfy a sort of concavity reversal of both Lyapunov's inequality,

due to Borell, while the Prekopa-Leindler inequality gives a reversal

of Holder. These inequalities are foundational in convex geometry,

give Renyi entropy comparisons in information theory, the Gaussian

log-Sobolev inequality, and more generally the HWI inequality in

optimal transport among other applications. An analogous theory has

been developing in the discrete setting. In this talk 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

counterexamples.