Iterated Entropy Derivatives and Binary Entropy Inequalities
Tanay Wakhare
TL;DR
The paper develops a comprehensive framework to prove the entropy inequality $\alpha_k H(x^k) \ge x^{k-1}H(x)$ for real exponents by deriving closed forms for the $(k+1)$-st derivative of $x^{k-r}H(x^r)$ in multiple representations, including an infinite series, a rational form, and a generalized-Stirling expansion. It reduces the real-$k$ case to a root-count problem for an explicit polynomial $p_{k,r}(x)$, enabling proofs for rational exponents (e.g., $k=3/2$) and suggesting a general method for tight inequalities of sums of polynomials times logarithms. The work connects combinatorial structures (generalized Stirling and Eulerian numbers) with entropy inequalities and provides explicit formulas for important special cases ($r=1$ and $r=k$), along with asymptotic behavior for $\alpha_k$ and a Lagrange-inversion–based description. This framework opens a path to proving entropy-type inequalities via controlled root analyses and may have broader implications for inequalities involving logarithms of polynomials in statistical mechanics and combinatorial optimization.
Abstract
We embark on a systematic study of the $(k+1)$-th derivative of $x^{k-r}H(x^r)$, where $H(x):=-x\log x-(1-x)\log(1-x)$ is the binary entropy and $k>r\geq 1$ are integers. Our motivation is the conjectural entropy inequality $α_k H(x^k)\geq x^{k-1}H(x)$, where $0<α_k<1$ is given by a functional equation. The $k=2$ case was the key technical tool driving recent breakthroughs on the union-closed sets conjecture. We express $ \frac{d^{k+1}}{dx^{k+1}}x^{k-r}H(x^r)$ as a rational function, an infinite series, and a sum over generalized Stirling numbers. This allows us to reduce the proof of the entropy inequality for real $k$ to showing that an associated polynomial has only two real roots in the interval $(0,1)$, which also allows us to prove the inequality for fractional exponents such as $k=3/2$. The proof suggests a new framework for proving tight inequalities for the sum of polynomials times the logarithms of polynomials, which converts the inequality into a statement about the real roots of a simpler associated polynomial.