Iterated Entropy Derivatives and Binary Entropy Inequalities
(2312.14743)Abstract
We embark on a systematic study of the $(k+1)$-th derivative of $x{k-r}H(xr)$, 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 $\alphak H(xk)\geq x{k-1}H(x)$, where $0<\alphak<1$ 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(xr)$ 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.
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