A Useful Inequality for the Binary Entropy Function

TL;DR

This paper presents a straightforward proof of a notable inequality for the binary entropy function, which has applications in Boolean formula lower bounds and combinatorial conjectures.

Contribution

It offers a simple differential calculus-based proof of a previously used inequality, enhancing understanding and potential applications.

Findings

Proved a key inequality for the binary entropy function.

Demonstrated the inequality's relevance to Boolean formulas and set conjectures.

Simplified the proof technique using basic calculus.

Abstract

We provide a simple proof of a curious inequality for the binary entropy function, an inequality that has been used in two different contexts. In the 1980's, Boppana used this entropy inequality to prove lower bounds on Boolean formulas. More recently, the inequality was used to achieve major progress on Frankl's union-closed sets conjecture. Our proof of the entropy inequality uses basic differential calculus.