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

TL;DR

This paper proves a reversed form of Lyapunov's inequality for monotone log-concave sequences, leading to new bounds in information theory, and explores the limitations of these results in symmetric cases.

Contribution

It establishes a reversed Lyapunov inequality for log-concave sequences and derives new information theoretic bounds, disproving a strengthened conjecture and extending results to symmetric settings.

Findings

Reversal of Lyapunov's inequality for monotone log-concave sequences.

Sharp bounds for varentropy, Rényi entropies, and information concentration.

Counterexamples showing limitations in symmetric cases.

Abstract

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.