A generalization of Boppana's entropy inequality
Boon Suan Ho
TL;DR
The paper generalizes Boppana's entropy inequality to real parameters $k>1$, proving the existence of a unique $0<\alpha_k<1$ satisfying $\alpha_k(1+\alpha_k)^{k-1}=1$ such that $\alpha_k h(x^k)\ge x^{k-1} h(x)$ for all $0\le x\le1$, with equality at $x=0$, $x=1/(1+\alpha_k)$, or $x=1$. This yields an analogue of the union-closed sets conjecture for approximate $k$-union closed set systems. The proof centers on analyzing $q(x)= x^{k-1} h(x)/h(x^k)$ and uses the auxiliary function $U(x)= \log x\,\log(1-x)/h(x)$ to pinpoint the unique critical point via $U(x)=U(x^k)$, which forces $x^k=1-x$ and leads to $x=1/(1+\alpha_k)$. The result is formalized in Lean 4 and connects to Yuster's conjecture on approximate $k$-union closed systems, contributing to the broader approach to the union-closed sets problem.
Abstract
In recent progress on the union-closed sets conjecture, a key lemma has been Boppana's entropy inequality: $h(x^2)\geφxh(x)$, where $φ=(1+\sqrt5)/2$ and $h(x)=-x\log x-(1-x)\log(1-x)$. In this note, we prove that the generalized inequality $α_kh(x^k)\ge x^{k-1}h(x)$, first conjectured by Yuster, holds for real $k>1$, where $α_k$ is the unique positive solution to $x(1+x)^{k-1}=1$. This implies an analogue of the union-closed sets conjecture for approximate $k$-union closed set systems. We also formalize our proof in Lean 4.