New Fractional Hermite–Hadamard–Mercer Inequalities for Harmonically Convex Function
In 2003, Mercer presented an interesting variation of Jensen’s inequality called Jensen–Mercer inequality for convex function. In the present paper, by employing harmonically convex function, we introduce analogous versions of Hermite–Hadamard inequalities of the Jensen–Mercer type via fractional in...
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| Published in: | Journal of function spaces Vol. 2021; pp. 1 - 11 |
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| Main Authors: | , , , , |
| Format: | Journal Article |
| Language: | English |
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New York
Hindawi
2021
John Wiley & Sons, Inc Wiley |
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| ISSN: | 2314-8896, 2314-8888 |
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| Abstract | In 2003, Mercer presented an interesting variation of Jensen’s inequality called Jensen–Mercer inequality for convex function. In the present paper, by employing harmonically convex function, we introduce analogous versions of Hermite–Hadamard inequalities of the Jensen–Mercer type via fractional integrals. As a result, we introduce several related fractional inequalities connected with the right and left differences of obtained new inequalities for differentiable harmonically convex mappings. As an application viewpoint, new estimates regarding hypergeometric functions and special means of real numbers are exemplified to determine the pertinence and validity of the suggested scheme. Our results presented here provide extensions of others given in the literature. The results proved in this paper may stimulate further research in this fascinating area. |
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| AbstractList | In 2003, Mercer presented an interesting variation of Jensen’s inequality called Jensen–Mercer inequality for convex function. In the present paper, by employing harmonically convex function, we introduce analogous versions of Hermite–Hadamard inequalities of the Jensen–Mercer type via fractional integrals. As a result, we introduce several related fractional inequalities connected with the right and left differences of obtained new inequalities for differentiable harmonically convex mappings. As an application viewpoint, new estimates regarding hypergeometric functions and special means of real numbers are exemplified to determine the pertinence and validity of the suggested scheme. Our results presented here provide extensions of others given in the literature. The results proved in this paper may stimulate further research in this fascinating area. |
| Author | Khan, Khuram Ali Yousaf, Saba Butt, Saad Ihsan Asghar, Atifa Moradi, Hamid Reza |
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| Cites_doi | 10.1016/j.jmaa.2007.02.016 10.1002/mma.6944 10.3934/math.2020412 10.1016/j.aej.2020.06.040 10.1515/phys-2020-0114 10.1186/1029-242x-2013-325 10.1002/num.22521 10.22436/jmcs.022.03.07 10.7153/jmi-2020-14-24 10.15672/HJMS.2014437519 10.1186/s13662-020-03093-y 10.1155/2020/8928691 10.1186/1029-242X-2013-527 10.7153/jmi-2020-14-17 10.1155/2020/7061549 10.1002/mma.6869 10.13001/1081-3810.1684 |
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| Copyright | Copyright © 2021 Saad Ihsan Butt et al. Copyright © 2021 Saad Ihsan Butt et al. This is an open access article distributed under the Creative Commons Attribution License (the “License”), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. https://creativecommons.org/licenses/by/4.0 |
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| References | 22 S. Dimitrijev (15) 1997 24 25 26 I. A. Baloch (18) 2020; 5 M. A. Khan (3) 2021; 44 S. S. Dragomir (5) 2000 F. Al-Azemi (13) 2015; 9 A. A. Kilbas (20) 2006 A. M. D. Mercer (6) 2003; 4 H. R. Moradi (8) 2007; 14 A. P. Prudnikov (21) 1981 N. Mehmood (1) 2007; 14 S. I. Butt (2) 2021; 44 S. Rashid (16) 2020; 18 10 11 D. S. Mitrinović (4) 1993 12 M. A. Noor (14) 2017 S. S. Dragomir (17) 2020; 27 H. H. Chu (23) 2020; 59 19 Í. Íşcan (27) 2014; 238 7 9 |
| References_xml | – ident: 11 doi: 10.1016/j.jmaa.2007.02.016 – volume: 44 start-page: 3324 issue: 5 year: 2021 ident: 3 article-title: A novel approach to the Jensen gap through Taylor’s theorem publication-title: Mathematical Methods in the Applied Sciences doi: 10.1002/mma.6944 – volume: 5 start-page: 6404 issue: 6 year: 2020 ident: 18 article-title: A varient of Jensen-type inequality and related results for harmonic convex functions publication-title: Aims Mathematics doi: 10.3934/math.2020412 – volume: 27 start-page: 103 year: 2020 ident: 17 article-title: Inequalities of Jensen type for HA-convex functions publication-title: Analele Universitatii din Oradea Fascicola Matematica – volume: 59 start-page: 3079 issue: 5 year: 2020 ident: 23 article-title: New fractional estimates for Hermite–Hadamard–Mercer’s type inequalities publication-title: Alexandria Engineering Journal doi: 10.1016/j.aej.2020.06.040 – volume-title: Theory and Applications of Fractional Differential Equations year: 2006 ident: 20 – volume: 18 start-page: 478 issue: 1 year: 2020 ident: 16 article-title: Some new extensions for fractional integral operator having exponential in the kernel and their applications in physical systems publication-title: Open Physics doi: 10.1515/phys-2020-0114 – ident: 22 doi: 10.1186/1029-242x-2013-325 – ident: 26 doi: 10.1002/num.22521 – volume-title: Selected Topics on Hermite-Hadamard Inequalities and Applications year: 2000 ident: 5 – volume-title: Integral and Series. Elementary Functions, vol. 1 year: 1981 ident: 21 – ident: 19 doi: 10.22436/jmcs.022.03.07 – volume-title: Advanced Convex Analysis and Optimization year: 2017 ident: 14 – volume: 14 start-page: 377 issue: 2 year: 2007 ident: 8 article-title: Improvement and generalization of some Jensen-Mercer-type inequalities publication-title: Journal of Mathematical Inequalities doi: 10.7153/jmi-2020-14-24 – volume-title: Effective Mass in Semiconductors year: 1997 ident: 15 – volume: 238 start-page: 236 issue: 1 year: 2014 ident: 27 article-title: Hermite-Hadamard type inequalities for harmonically convex functions via fractional integral publication-title: Applied Mathematics and Computation – ident: 12 doi: 10.15672/HJMS.2014437519 – volume: 9 start-page: 1 year: 2015 ident: 13 article-title: Asian options with harmonic average publication-title: Applied Mathematics and Information Sciences – ident: 24 doi: 10.1186/s13662-020-03093-y – volume-title: Classical and New Inequalities in Analysis, Mathematics and Its Applications (East European Series), 61 year: 1993 ident: 4 – ident: 9 doi: 10.1155/2020/8928691 – ident: 10 doi: 10.1186/1029-242X-2013-527 – volume: 14 start-page: 249 issue: 1 year: 2007 ident: 1 article-title: Generalizations of cyclic refinements of Jensen’s inequality by Lidstone’s polynomial with applications in information theory publication-title: Journal of Mathematical Inequalities doi: 10.7153/jmi-2020-14-17 – ident: 25 doi: 10.1155/2020/7061549 – volume: 4 issue: 4 year: 2003 ident: 6 article-title: A variant of Jensens inequality publication-title: Journal of Inequalities in Pure and Applied Mathematics – volume: 44 start-page: 1664 issue: 2 year: 2021 ident: 2 article-title: Jensen-Grüss inequality and its applications for the Zipf-Mandelbrot law publication-title: Mathematical Methods in the Applied Sciences doi: 10.1002/mma.6869 – ident: 7 doi: 10.13001/1081-3810.1684 |
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| SubjectTerms | Calculus Convex analysis Fractional calculus Hypergeometric functions Inequalities Information theory Integrals Random variables Real numbers |
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| Title | New Fractional Hermite–Hadamard–Mercer Inequalities for Harmonically Convex Function |
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