Inequalities, identities, and bounds for divided differences of the exponential function
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914089620471808 |
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| author | Zeng, Qiulin Ezzell, Nicholas Babakhani, Arman Hen, Itay Barash, Lev |
| author_facet | Zeng, Qiulin Ezzell, Nicholas Babakhani, Arman Hen, Itay Barash, Lev |
| contents | Let $\exp[x_0,x_1,\dots,x_n]$ denote the divided difference of the exponential function.
(i) We prove that exponential divided differences are log-submodular.
(ii) We establish the four-point inequality $ \exp[a,a,b,c]\,\exp[d,d,b,c]+\exp[b,b,a,d]\,\exp[c,c,a,d]-\exp[a,b,c,d]^2 \ge 0 $ for all $ a,b,c,d \in \mathbb{R} $.
(iii) We obtain sharp two-sided bounds for $\exp[x_0,\dots,x_n]$ at fixed mean and variance; as a consequence, we derive their large-input asymptotics.
(iv) We present closed-form identities for divided differences of the exponential function, including a convolution identity and summation formulas for repeated arguments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10724 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inequalities, identities, and bounds for divided differences of the exponential function Zeng, Qiulin Ezzell, Nicholas Babakhani, Arman Hen, Itay Barash, Lev Classical Analysis and ODEs Numerical Analysis Let $\exp[x_0,x_1,\dots,x_n]$ denote the divided difference of the exponential function. (i) We prove that exponential divided differences are log-submodular. (ii) We establish the four-point inequality $ \exp[a,a,b,c]\,\exp[d,d,b,c]+\exp[b,b,a,d]\,\exp[c,c,a,d]-\exp[a,b,c,d]^2 \ge 0 $ for all $ a,b,c,d \in \mathbb{R} $. (iii) We obtain sharp two-sided bounds for $\exp[x_0,\dots,x_n]$ at fixed mean and variance; as a consequence, we derive their large-input asymptotics. (iv) We present closed-form identities for divided differences of the exponential function, including a convolution identity and summation formulas for repeated arguments. |
| title | Inequalities, identities, and bounds for divided differences of the exponential function |
| topic | Classical Analysis and ODEs Numerical Analysis |
| url | https://arxiv.org/abs/2510.10724 |