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

@article{Melbourne2021ADC, title={A discrete complement of Lyapunov's inequality and its information theoretic consequences}, author={James Melbourne and Gerardo Palafox-Castillo}, journal={ArXiv}, year={2021}, volume={abs/2111.06997} }

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, Rényi entropies, and the concentration of information of monotone logconcave random variables. Moreover, the majorization approach…

## 2 Citations

### On a Conjecture of Feige for Discrete Log-Concave Distributions

- Mathematics
- 2022

A remarkable conjecture of Feige (2006) asserts that for any collection of n independent non-negative random variables X 1 , X 2 , . . . , X n , each with expectation at most 1, P ( X < E [ X ] + 1)…

### Moments, Concentration, and Entropy of Log-Concave Distributions

- EconomicsArXiv
- 2022

We utilize and extend a simple and classical mechanism, combining log-concavity and majorization in the convex order to derive moments, concentration, and entropy inequalities for log-concave random…

## References

SHOWING 1-10 OF 45 REFERENCES

### Monotonicity, Thinning, and Discrete Versions of the Entropy Power Inequality

- Computer ScienceIEEE Transactions on Information Theory
- 2010

A stronger version of concavity of entropy is proved, which implies a strengthened form of Shannon's discrete EPI, which gives a sharp bound on how the entropy of ultra log-concave random variables behaves on thinning.

### Displacement convexity of entropy and related inequalities on graphs

- Mathematics
- 2012

We introduce the notion of an interpolating path on the set of probability measures on finite graphs. Using this notion, we first prove a displacement convexity property of entropy along such a path…

### Rényi Entropy Power Inequalities via Normal Transport and Rotation

- Computer ScienceEntropy
- 2018

A comprehensive framework for deriving various EPIs for the Rényi entropy is presented that uses transport arguments from normal densities and a change of variable by rotation, and a simple transportation proof of a sharp varentropy bound is obtained.

### Further Investigations of Rényi Entropy Power Inequalities and an Entropic Characterization of s-Concave Densities

- Mathematics, Computer ScienceLecture Notes in Mathematics
- 2020

The role of convexity in Renyi entropy power inequalities is investigated and the convergence in the Central Limit Theorem for Renyi entropies of order r ∈ (0, 1) for log-concave densities and for compactly supported, spherically symmetric and unimodal densities is established.

### Rényi entropy power inequality and a reverse

- Computer ScienceArXiv
- 2017

A refinement of the R\'enyi Entropy Power Inequality recently obtained inBM16 is presented, and a conjecture in BNT15, MMX16 in two cases is confirmed, which largely follows the approach in DCT91 of employing Young's convolution inequalities with sharp constants.

### Geometric and functional inequalities for log-concave probability sequences

- Mathematics
- 2020

We investigate various geometric and functional inequalities for the class of log-concave probability sequences. We prove dilation inequalities for log-concave probability measures on the integers. A…

### Transport Proofs of some discrete variants of the Prékopa-Leindler inequality

- MathematicsANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
- 2021

We give a transport proof of a discrete version of the displacement convexity of entropy on integers (Z), and get, as a consequence, two discrete forms of the Prekopa-Leindler Inequality : the Four…

### On the problem of reversibility of the entropy power inequality

- MathematicsArXiv
- 2011

It is proved that reversibility is impossible over the whole class of convex probability distributions, and related phenomena for identically distributed summands are discussed.

### A Remark on discrete Brunn-Minkowski type inequalities via transportation of measure

- Mathematics
- 2020

We give an alternative proof for discrete Brunn-Minkowski type inequalities, recently obtained by Halikias, Klartag and the author. This proof also implies stronger weighted versions of these…

### Poisson processes and a log-concave Bernstein theorem

- MathematicsStudia Mathematica
- 2019

We discuss interplays between log-concave functions and log-concave sequences. We prove a Bernstein-type theorem, which characterizes the Laplace transform of log-concave measures on the half-line in…