Some Hadamard–Fejér Type Inequalities for LR-Convex Interval-Valued Functions
The purpose of this study is to introduce the new class of Hermite–Hadamard inequality for LR-convex interval-valued functions known as LR-interval Hermite–Hadamard inequality, by means of pseudo-order relation ( <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" dis...
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2021-12-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/6/1/6 |
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| author | Muhammad Bilal Khan Savin Treanțǎ Mohamed S. Soliman Kamsing Nonlaopon Hatim Ghazi Zaini |
| author_facet | Muhammad Bilal Khan Savin Treanțǎ Mohamed S. Soliman Kamsing Nonlaopon Hatim Ghazi Zaini |
| author_sort | Muhammad Bilal Khan |
| collection | DOAJ |
| description | The purpose of this study is to introduce the new class of Hermite–Hadamard inequality for LR-convex interval-valued functions known as LR-interval Hermite–Hadamard inequality, by means of pseudo-order relation ( <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>≤</mo><mi>p</mi></msub></mrow></semantics></math></inline-formula> ). This order relation is defined on interval space. We have proved that if the interval-valued function is LR-convex then the inclusion relation “ <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>⊆</mo></semantics></math></inline-formula> ” coincident to pseudo-order relation “ <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>≤</mo><mi>p</mi></msub></mrow></semantics></math></inline-formula> ” under some suitable conditions. Moreover, the interval Hermite–Hadamard–Fejér inequality is also derived for LR-convex interval-valued functions. These inequalities also generalize some new and known results. Useful examples that verify the applicability of the theory developed in this study are presented. The concepts and techniques of this paper may be a starting point for further research in this area. |
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| id | doaj.art-03061f107d0c413482dd3c85ea70bc0b |
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| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T01:27:22Z |
| publishDate | 2021-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-03061f107d0c413482dd3c85ea70bc0b2023-11-23T13:48:31ZengMDPI AGFractal and Fractional2504-31102021-12-0161610.3390/fractalfract6010006Some Hadamard–Fejér Type Inequalities for LR-Convex Interval-Valued FunctionsMuhammad Bilal Khan0Savin Treanțǎ1Mohamed S. Soliman2Kamsing Nonlaopon3Hatim Ghazi Zaini4Department of Mathematics, COMSATS University Islamabad, Islamabad 44000, PakistanDepartment of Applied Mathematics, University Politehnica of Bucharest, 060042 Bucharest, RomaniaDepartment of Electrical Engineering, College of Engineering, Taif University, P.O. Box 11099, Taif 21944, Saudi ArabiaDepartment of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, ThailandDepartment of Computer Science, College of Computers and Information Technology, Taif University, P.O. Box 11099, Taif 21944, Saudi ArabiaThe purpose of this study is to introduce the new class of Hermite–Hadamard inequality for LR-convex interval-valued functions known as LR-interval Hermite–Hadamard inequality, by means of pseudo-order relation ( <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>≤</mo><mi>p</mi></msub></mrow></semantics></math></inline-formula> ). This order relation is defined on interval space. We have proved that if the interval-valued function is LR-convex then the inclusion relation “ <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>⊆</mo></semantics></math></inline-formula> ” coincident to pseudo-order relation “ <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>≤</mo><mi>p</mi></msub></mrow></semantics></math></inline-formula> ” under some suitable conditions. Moreover, the interval Hermite–Hadamard–Fejér inequality is also derived for LR-convex interval-valued functions. These inequalities also generalize some new and known results. Useful examples that verify the applicability of the theory developed in this study are presented. The concepts and techniques of this paper may be a starting point for further research in this area.https://www.mdpi.com/2504-3110/6/1/6interval-valued functionRiemann integralLR-convex interval-valued functioninterval Hermite–Hadamard inequalityinterval Hermite–Hadamard–Fejér inequality |
| spellingShingle | Muhammad Bilal Khan Savin Treanțǎ Mohamed S. Soliman Kamsing Nonlaopon Hatim Ghazi Zaini Some Hadamard–Fejér Type Inequalities for LR-Convex Interval-Valued Functions Fractal and Fractional interval-valued function Riemann integral LR-convex interval-valued function interval Hermite–Hadamard inequality interval Hermite–Hadamard–Fejér inequality |
| title | Some Hadamard–Fejér Type Inequalities for LR-Convex Interval-Valued Functions |
| title_full | Some Hadamard–Fejér Type Inequalities for LR-Convex Interval-Valued Functions |
| title_fullStr | Some Hadamard–Fejér Type Inequalities for LR-Convex Interval-Valued Functions |
| title_full_unstemmed | Some Hadamard–Fejér Type Inequalities for LR-Convex Interval-Valued Functions |
| title_short | Some Hadamard–Fejér Type Inequalities for LR-Convex Interval-Valued Functions |
| title_sort | some hadamard fejer type inequalities for lr convex interval valued functions |
| topic | interval-valued function Riemann integral LR-convex interval-valued function interval Hermite–Hadamard inequality interval Hermite–Hadamard–Fejér inequality |
| url | https://www.mdpi.com/2504-3110/6/1/6 |
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