Iterated entropy derivatives and binary entropy inequalities

We embark on a systematic study of the $(k+1)$-th derivative of $x^{k-r}H\left(x^r\right)$, where $H\left(x\right)≔-xlogx-(1-x)log(1-x)$ is the binary entropy and $k\ge r\ge 1$ are integers. Our motivation is the conjectural entropy inequality ${\alpha }_{k}H\left(x^k\right)\ge x^{k-1}H\left(x\right)$, where $0<{\alpha }_k<1$ 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 $\frac{d^{k+1}}{d{x}^{k+1}}{x}^{k-r}H\left(x^r\right)$ 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.