A generalization of Boppana's entropy inequality

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.