Iterated entropy derivatives and binary entropy inequalities

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 >= 1 are integers. Our motivation is the conjectural entropy inequality alpha H-k (x(k)) >= 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 d(k+1)/dx(k+r) 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. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.