A simple proof of Popoviciu's inequality
T. Popoviciu [5] has proved in 1965 the following inequality relating the values of a convex function \(f:I\rightarrow\mathbb{R}\) at the weighted arithmetic means of the subfamilies of a given family of points \(x_{1},...,x_{n}\in I\):\begin{align*}& \sum\limits_{1\leq i_{1}<\cdots <i_{p}...
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| Format: | Article |
| Language: | English |
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Publishing House of the Romanian Academy
2008-08-01
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| Series: | Journal of Numerical Analysis and Approximation Theory |
| Subjects: | |
| Online Access: | https://ictp.acad.ro/jnaat/journal/article/view/884 |
| _version_ | 1818474931044745216 |
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| author | Mihaly Bencze Florin Popovici |
| author_facet | Mihaly Bencze Florin Popovici |
| author_sort | Mihaly Bencze |
| collection | DOAJ |
| description | T. Popoviciu [5] has proved in 1965 the following inequality relating the values of a convex function \(f:I\rightarrow\mathbb{R}\) at the weighted arithmetic means of the subfamilies of a given family of points \(x_{1},...,x_{n}\in I\):\begin{align*}& \sum\limits_{1\leq i_{1}<\cdots <i_{p}\leq n}(\lambda_{i_{1}}+\cdots +\lambda _{i_{p}})\,f\left( \tfrac{\lambda_{i_{1}}x_{i_{1}}+\cdots +\lambda_{i_{p}}x_{i_{p}}}{\lambda _{i_{1}}+\cdots +\lambda _{i_{p}}}\right) \\& \leq \tbinom{n-2}{p-2}\left[\tfrac{n-p}{p-1}\,\sum\limits_{i=1}^{n}\,\lambda_{i}\,f(x_{i})+\left( \sum\limits_{i=1}^{n}\,\lambda _{i}\right) \,f\left( \tfrac{\lambda _{1}x_{1}+\cdots +\lambda _{n}x_{n}}{\lambda _{1}+\cdots+\lambda _{n}}\right) \right] .\end{align*}Here \(n\geq 3,\) \(p\in \{2,...,n-1\}\) and \(\lambda _{1},...,\lambda_{n}\) are positive numbers (representing weights). The aim of this paper is to give a simple argument based on mathematical induction and a majorization lemma. |
| first_indexed | 2024-04-14T04:43:27Z |
| format | Article |
| id | doaj.art-3bddd3e9a5164cb785bd3b15526c2a22 |
| institution | Directory Open Access Journal |
| issn | 2457-6794 2501-059X |
| language | English |
| last_indexed | 2024-04-14T04:43:27Z |
| publishDate | 2008-08-01 |
| publisher | Publishing House of the Romanian Academy |
| record_format | Article |
| series | Journal of Numerical Analysis and Approximation Theory |
| spelling | doaj.art-3bddd3e9a5164cb785bd3b15526c2a222022-12-22T02:11:33ZengPublishing House of the Romanian AcademyJournal of Numerical Analysis and Approximation Theory2457-67942501-059X2008-08-01372A simple proof of Popoviciu's inequalityMihaly Bencze0Florin Popovici1Aprily Lajos College, BraşovGrigore Moisil College, BraşovT. Popoviciu [5] has proved in 1965 the following inequality relating the values of a convex function \(f:I\rightarrow\mathbb{R}\) at the weighted arithmetic means of the subfamilies of a given family of points \(x_{1},...,x_{n}\in I\):\begin{align*}& \sum\limits_{1\leq i_{1}<\cdots <i_{p}\leq n}(\lambda_{i_{1}}+\cdots +\lambda _{i_{p}})\,f\left( \tfrac{\lambda_{i_{1}}x_{i_{1}}+\cdots +\lambda_{i_{p}}x_{i_{p}}}{\lambda _{i_{1}}+\cdots +\lambda _{i_{p}}}\right) \\& \leq \tbinom{n-2}{p-2}\left[\tfrac{n-p}{p-1}\,\sum\limits_{i=1}^{n}\,\lambda_{i}\,f(x_{i})+\left( \sum\limits_{i=1}^{n}\,\lambda _{i}\right) \,f\left( \tfrac{\lambda _{1}x_{1}+\cdots +\lambda _{n}x_{n}}{\lambda _{1}+\cdots+\lambda _{n}}\right) \right] .\end{align*}Here \(n\geq 3,\) \(p\in \{2,...,n-1\}\) and \(\lambda _{1},...,\lambda_{n}\) are positive numbers (representing weights). The aim of this paper is to give a simple argument based on mathematical induction and a majorization lemma.https://ictp.acad.ro/jnaat/journal/article/view/884Popoviciu's inequalityconvex functionconvex combination |
| spellingShingle | Mihaly Bencze Florin Popovici A simple proof of Popoviciu's inequality Journal of Numerical Analysis and Approximation Theory Popoviciu's inequality convex function convex combination |
| title | A simple proof of Popoviciu's inequality |
| title_full | A simple proof of Popoviciu's inequality |
| title_fullStr | A simple proof of Popoviciu's inequality |
| title_full_unstemmed | A simple proof of Popoviciu's inequality |
| title_short | A simple proof of Popoviciu's inequality |
| title_sort | simple proof of popoviciu s inequality |
| topic | Popoviciu's inequality convex function convex combination |
| url | https://ictp.acad.ro/jnaat/journal/article/view/884 |
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