On generalization of refinement of Jensen’s inequality using Fink’s identity and Abel-Gontscharoff Green function
Abstract In this paper, we formulate new Abel-Gontscharoff type identities involving new Green functions for the ‘two-point right focal’ problem. We use Fink’s identity and a new Abel-Gontscharoff-type Green’s function for a ‘two-point right focal’ to generalize the refinement of Jensen’s inequality...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2017-10-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13660-017-1521-x |
| _version_ | 1819136515967549440 |
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| author | Tasadduq Niaz Khuram Ali Khan Josip Pečarić |
| author_facet | Tasadduq Niaz Khuram Ali Khan Josip Pečarić |
| author_sort | Tasadduq Niaz |
| collection | DOAJ |
| description | Abstract In this paper, we formulate new Abel-Gontscharoff type identities involving new Green functions for the ‘two-point right focal’ problem. We use Fink’s identity and a new Abel-Gontscharoff-type Green’s function for a ‘two-point right focal’ to generalize the refinement of Jensen’s inequality given in (Horváth and Pečarić in Math. Inequal. Appl. 14: 777-791, 2011) from convex function to higher order convex function. Also we formulate the monotonicity of the linear functional obtained from these identities using the recent theory of inequalities for n-convex function at a point. Further we give the bounds for the identities related to the generalization of the refinement of Jensen’s inequality using inequalities for the Cebyšev functional. Some results relating to the Grüss and Ostrowski-type inequalities are constructed. |
| first_indexed | 2024-12-22T10:36:13Z |
| format | Article |
| id | doaj.art-9424ea5be9c44411b8bd28863018f9ae |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-22T10:36:13Z |
| publishDate | 2017-10-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-9424ea5be9c44411b8bd28863018f9ae2022-12-21T18:29:10ZengSpringerOpenJournal of Inequalities and Applications1029-242X2017-10-012017111210.1186/s13660-017-1521-xOn generalization of refinement of Jensen’s inequality using Fink’s identity and Abel-Gontscharoff Green functionTasadduq Niaz0Khuram Ali Khan1Josip Pečarić2Department of Mathematics, University of SargodhaDepartment of Mathematics, University of SargodhaRUDN UniversityAbstract In this paper, we formulate new Abel-Gontscharoff type identities involving new Green functions for the ‘two-point right focal’ problem. We use Fink’s identity and a new Abel-Gontscharoff-type Green’s function for a ‘two-point right focal’ to generalize the refinement of Jensen’s inequality given in (Horváth and Pečarić in Math. Inequal. Appl. 14: 777-791, 2011) from convex function to higher order convex function. Also we formulate the monotonicity of the linear functional obtained from these identities using the recent theory of inequalities for n-convex function at a point. Further we give the bounds for the identities related to the generalization of the refinement of Jensen’s inequality using inequalities for the Cebyšev functional. Some results relating to the Grüss and Ostrowski-type inequalities are constructed.http://link.springer.com/article/10.1186/s13660-017-1521-xconvex functionJensen’s inequalityFink’s identityAbel-Gontscharoff interpolating polynomialGreen function for ‘two-point right focal’ problem |
| spellingShingle | Tasadduq Niaz Khuram Ali Khan Josip Pečarić On generalization of refinement of Jensen’s inequality using Fink’s identity and Abel-Gontscharoff Green function Journal of Inequalities and Applications convex function Jensen’s inequality Fink’s identity Abel-Gontscharoff interpolating polynomial Green function for ‘two-point right focal’ problem |
| title | On generalization of refinement of Jensen’s inequality using Fink’s identity and Abel-Gontscharoff Green function |
| title_full | On generalization of refinement of Jensen’s inequality using Fink’s identity and Abel-Gontscharoff Green function |
| title_fullStr | On generalization of refinement of Jensen’s inequality using Fink’s identity and Abel-Gontscharoff Green function |
| title_full_unstemmed | On generalization of refinement of Jensen’s inequality using Fink’s identity and Abel-Gontscharoff Green function |
| title_short | On generalization of refinement of Jensen’s inequality using Fink’s identity and Abel-Gontscharoff Green function |
| title_sort | on generalization of refinement of jensen s inequality using fink s identity and abel gontscharoff green function |
| topic | convex function Jensen’s inequality Fink’s identity Abel-Gontscharoff interpolating polynomial Green function for ‘two-point right focal’ problem |
| url | http://link.springer.com/article/10.1186/s13660-017-1521-x |
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