On the Generalization of Weinberger's Inequality with Alternating Signs
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911961681231872 |
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| author | Yadeta, Hailu Bikila |
| author_facet | Yadeta, Hailu Bikila |
| contents | For given set of $m$ positive numbers satisfying the conditions: $$ a_1 \geq a_2 \geq , ... \geq a_m \geq 0, $$ the inequality $$ \sum_{s=1}^{m} (-1)^{s-1}a^r_s \geq \left[ \sum_{s=1}^{m} (-1)^{s-1}a_s\right]^r, \quad r > 1, $$ was proved by H. Weinberger. The generalization of Weinberger's result takes the form $$ \sum_{s=1}^{m} (-1)^{s-1}f(a_s) \geq f\left( \sum_{s=1}^{m} (-1)^{s-1}a_s\right), $$ where $f$ is a convex function satisfying the condition $f(0)\leq 0 $. The condition $f(0)\geq 0 $ in the generalization proposed by Bellman was corrected by Olkin as $f(0) \leq 0 $. Bellman gave only a graphical proof for differentiable convex functions. In this paper, we give a mathematical proof for the generalized inequality including the importance of the condition $f(0)\leq 0$. We introduce a set $\mathcal{W}$ of functions so that functions in the intersection of $\mathcal{W}$ and the set of all convex functions are the ones that are desirable in the generalization. In addition, we give a proof of Szegö's inequality which applies to sums with odd number of terms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05496 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Generalization of Weinberger's Inequality with Alternating Signs Yadeta, Hailu Bikila Classical Analysis and ODEs For given set of $m$ positive numbers satisfying the conditions: $$ a_1 \geq a_2 \geq , ... \geq a_m \geq 0, $$ the inequality $$ \sum_{s=1}^{m} (-1)^{s-1}a^r_s \geq \left[ \sum_{s=1}^{m} (-1)^{s-1}a_s\right]^r, \quad r > 1, $$ was proved by H. Weinberger. The generalization of Weinberger's result takes the form $$ \sum_{s=1}^{m} (-1)^{s-1}f(a_s) \geq f\left( \sum_{s=1}^{m} (-1)^{s-1}a_s\right), $$ where $f$ is a convex function satisfying the condition $f(0)\leq 0 $. The condition $f(0)\geq 0 $ in the generalization proposed by Bellman was corrected by Olkin as $f(0) \leq 0 $. Bellman gave only a graphical proof for differentiable convex functions. In this paper, we give a mathematical proof for the generalized inequality including the importance of the condition $f(0)\leq 0$. We introduce a set $\mathcal{W}$ of functions so that functions in the intersection of $\mathcal{W}$ and the set of all convex functions are the ones that are desirable in the generalization. In addition, we give a proof of Szegö's inequality which applies to sums with odd number of terms. |
| title | On the Generalization of Weinberger's Inequality with Alternating Signs |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2407.05496 |