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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | https://arxiv.org/abs/2312.14743 |
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| _version_ | 1866915101400891392 |
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| author | Wakhare, Tanay |
| author_facet | Wakhare, Tanay |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_14743 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Iterated Entropy Derivatives and Binary Entropy Inequalities Wakhare, Tanay Information Theory Combinatorics Number Theory 94A17, 26C10, 11B65, 05A10 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. |
| title | Iterated Entropy Derivatives and Binary Entropy Inequalities |
| topic | Information Theory Combinatorics Number Theory 94A17, 26C10, 11B65, 05A10 |
| url | https://arxiv.org/abs/2312.14743 |