Sharp inequalities for symmetric polynomials, Hunter's conjecture, and moments of exponential random variables
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909959835353088 |
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| author | Brazitikos, Silouanos Pandis, Christos |
| author_facet | Brazitikos, Silouanos Pandis, Christos |
| contents | We prove Hunter's conjecture on complete homogeneous symmetric polynomials. For even $n$ and every integer $k\geq 1$, we show that under the constraint $\sum_{i=1}^n a_i^2=1$ the global minimum of the even-degree polynomial $h_{2k}(a_1,\dots,a_n)$ is attained precisely at the half-plus/half-minus vector and we compute the optimal value in closed form. The proof combines algebraic properties of $h_{2k}$ with the probabilistic representation $k!\,h_k(a)=\mathbb{E}(\sum_{i=1}^n a_iX_i)^k$, where $X_1,\dots,X_n$ are i.i.d. standard exponential random variables with density $e^{-x}1_{x>0}$ and a combinatorial identity. This viewpoint further yields sharp upper and lower bounds for $\mathbb{E}|\sum_{i=1}^n a_iX_i|^{q}$ under natural constraints on the coefficients, including the spherical constraint $\sum a_i^2=1$ combined with the non-negative regime $a_i\ge0$, or the centred regime $\sum a_i=0$. Moreover, we determine the exact minimum of $h_{2k}$ on the $\ell_\infty$-sphere $S_\infty = \{a \in \mathbb{R}^n : \|a\|_\infty = 1\}$, which yields sharp norm comparison inequalities between the matrix norms induced by complete homogeneous symmetric polynomials and the classical operator and Schatten norms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_12254 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp inequalities for symmetric polynomials, Hunter's conjecture, and moments of exponential random variables Brazitikos, Silouanos Pandis, Christos Probability Functional Analysis Primary 05E05, 60E15, 26D15, Secondary 15A60, 52A40 We prove Hunter's conjecture on complete homogeneous symmetric polynomials. For even $n$ and every integer $k\geq 1$, we show that under the constraint $\sum_{i=1}^n a_i^2=1$ the global minimum of the even-degree polynomial $h_{2k}(a_1,\dots,a_n)$ is attained precisely at the half-plus/half-minus vector and we compute the optimal value in closed form. The proof combines algebraic properties of $h_{2k}$ with the probabilistic representation $k!\,h_k(a)=\mathbb{E}(\sum_{i=1}^n a_iX_i)^k$, where $X_1,\dots,X_n$ are i.i.d. standard exponential random variables with density $e^{-x}1_{x>0}$ and a combinatorial identity. This viewpoint further yields sharp upper and lower bounds for $\mathbb{E}|\sum_{i=1}^n a_iX_i|^{q}$ under natural constraints on the coefficients, including the spherical constraint $\sum a_i^2=1$ combined with the non-negative regime $a_i\ge0$, or the centred regime $\sum a_i=0$. Moreover, we determine the exact minimum of $h_{2k}$ on the $\ell_\infty$-sphere $S_\infty = \{a \in \mathbb{R}^n : \|a\|_\infty = 1\}$, which yields sharp norm comparison inequalities between the matrix norms induced by complete homogeneous symmetric polynomials and the classical operator and Schatten norms. |
| title | Sharp inequalities for symmetric polynomials, Hunter's conjecture, and moments of exponential random variables |
| topic | Probability Functional Analysis Primary 05E05, 60E15, 26D15, Secondary 15A60, 52A40 |
| url | https://arxiv.org/abs/2512.12254 |