The Choquet integral of log-convex functions
Abstract In this paper we investigate the upper bound and the lower bound of the Choquet integral for log-convex functions. Firstly, for a monotone log-convex function, we state the similar Hadamard inequality of the Choquet integral in the framework of distorted measure. Secondly, we estimate the u...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2018-08-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13660-018-1803-y |
| _version_ | 1818544906698752000 |
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| author | Hongxia Wang |
| author_facet | Hongxia Wang |
| author_sort | Hongxia Wang |
| collection | DOAJ |
| description | Abstract In this paper we investigate the upper bound and the lower bound of the Choquet integral for log-convex functions. Firstly, for a monotone log-convex function, we state the similar Hadamard inequality of the Choquet integral in the framework of distorted measure. Secondly, we estimate the upper bound of the Choquet integral for a general log-convex function, respectively, in the case of distorted Lebesgue measure and in the non-additive measure. Finally, we present Jensen’s inequality of the Choquet integral for log-convex functions, which can be used to estimate the lower bound of this kind when the non-additive measure is concave. We provide some examples in the framework of the distorted Lebesgue measure to illustrate all the results. |
| first_indexed | 2024-12-11T22:54:34Z |
| format | Article |
| id | doaj.art-988c0fc07eb04284843626ad0acf37ff |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-11T22:54:34Z |
| publishDate | 2018-08-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-988c0fc07eb04284843626ad0acf37ff2022-12-22T00:47:17ZengSpringerOpenJournal of Inequalities and Applications1029-242X2018-08-012018111710.1186/s13660-018-1803-yThe Choquet integral of log-convex functionsHongxia Wang0College of Statistics, Henan University of Economics and LawAbstract In this paper we investigate the upper bound and the lower bound of the Choquet integral for log-convex functions. Firstly, for a monotone log-convex function, we state the similar Hadamard inequality of the Choquet integral in the framework of distorted measure. Secondly, we estimate the upper bound of the Choquet integral for a general log-convex function, respectively, in the case of distorted Lebesgue measure and in the non-additive measure. Finally, we present Jensen’s inequality of the Choquet integral for log-convex functions, which can be used to estimate the lower bound of this kind when the non-additive measure is concave. We provide some examples in the framework of the distorted Lebesgue measure to illustrate all the results.http://link.springer.com/article/10.1186/s13660-018-1803-yChoquet integralLog-convex functionInequality |
| spellingShingle | Hongxia Wang The Choquet integral of log-convex functions Journal of Inequalities and Applications Choquet integral Log-convex function Inequality |
| title | The Choquet integral of log-convex functions |
| title_full | The Choquet integral of log-convex functions |
| title_fullStr | The Choquet integral of log-convex functions |
| title_full_unstemmed | The Choquet integral of log-convex functions |
| title_short | The Choquet integral of log-convex functions |
| title_sort | choquet integral of log convex functions |
| topic | Choquet integral Log-convex function Inequality |
| url | http://link.springer.com/article/10.1186/s13660-018-1803-y |
| work_keys_str_mv | AT hongxiawang thechoquetintegraloflogconvexfunctions AT hongxiawang choquetintegraloflogconvexfunctions |