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 $\alpha_k H(x^k)\geq x^{k-1}H(x)$, 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}}{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.

翻译：我们系统研究了 $(k+1)$ 阶导数 $x^{k-r}H(x^r)$，其中 $H(x):=-x\log x-(1-x)\log(1-x)$ 为二元熵，$k>r\geq 1$ 为整数。我们的动机源于猜想中的熵不等式 $\alpha_k H(x^k)\geq x^{k-1}H(x)$，其中 $0<\alpha_k<1$ 由泛函方程给出。$k=2$ 情形是近期突破并集封闭猜想的关键技术工具。我们将 $\frac{d^{k+1}}{dx^{k+1}}x^{k-r}H(x^r)$ 表示为有理函数、无穷级数以及广义 Stirling 数的求和形式。这使我们能够将该熵不等式在实数 $k$ 下的证明归结为证明一个关联多项式在区间 $(0,1)$ 内仅有两个实根，并由此证明分数指数（如 $k=3/2$）情形下的不等式。该证明提出了一种新框架，用于证明多项式与对数多项式乘积之和的紧致不等式，其核心是将原不等式转化为关于一个更简单关联多项式实根的性质断言。