On Riemann–Liouville Integral via Strongly Modified (<i>h</i>,<i>m</i>)-Convex Functions
The generalization of strongly convex and strongly <i>m</i>-convex functions is presented in this paper. We began by proving the properties of a strongly modified <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semanti...
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MDPI AG
2024-11-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/8/12/680 |
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| author | Ali N. A. Koam Ammara Nosheen Khuram Ali Khan Mudassir Hussain Bukhari Ali Ahmad Maryam Salem Alatawi |
| author_facet | Ali N. A. Koam Ammara Nosheen Khuram Ali Khan Mudassir Hussain Bukhari Ali Ahmad Maryam Salem Alatawi |
| author_sort | Ali N. A. Koam |
| collection | DOAJ |
| description | The generalization of strongly convex and strongly <i>m</i>-convex functions is presented in this paper. We began by proving the properties of a strongly modified <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>h</mi><mo>,</mo><mi>m</mi><mo>)</mo></mrow></semantics></math></inline-formula>-convex function. The Schur inequality and the Hermite–Hadamard (H-H) inequalities are proved for the proposed class. Moreover, H-H inequalities are also proved in the context of Riemann–Liouville (R-L) integrals. Some examples and graphs are also presented in order to show the existence of this newly defined class. |
| format | Article |
| id | doaj-art-e3006eb3cbb641dcac22d422d7eb2c8f |
| institution | DOAJ |
| issn | 2504-3110 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-e3006eb3cbb641dcac22d422d7eb2c8f2025-08-20T02:53:38ZengMDPI AGFractal and Fractional2504-31102024-11-0181268010.3390/fractalfract8120680On Riemann–Liouville Integral via Strongly Modified (<i>h</i>,<i>m</i>)-Convex FunctionsAli N. A. Koam0Ammara Nosheen1Khuram Ali Khan2Mudassir Hussain Bukhari3Ali Ahmad4Maryam Salem Alatawi5Department of Mathematics, College of Science, Jazan University, Jazan 45142, Saudi ArabiaDepartment of Mathematics, University of Sargodha, Sargodha 40100, PakistanDepartment of Mathematics, University of Sargodha, Sargodha 40100, PakistanDepartment of Mathematics, University of Sargodha, Sargodha 40100, PakistanDepartment of Computer Science, College of Engineering and Computer Science, Jazan University, Jazan 45142, Saudi ArabiaDepartment of Mathematics, Faculty of Sciences, University of Tabuk, Tabuk 71491, Saudi ArabiaThe generalization of strongly convex and strongly <i>m</i>-convex functions is presented in this paper. We began by proving the properties of a strongly modified <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>h</mi><mo>,</mo><mi>m</mi><mo>)</mo></mrow></semantics></math></inline-formula>-convex function. The Schur inequality and the Hermite–Hadamard (H-H) inequalities are proved for the proposed class. Moreover, H-H inequalities are also proved in the context of Riemann–Liouville (R-L) integrals. Some examples and graphs are also presented in order to show the existence of this newly defined class.https://www.mdpi.com/2504-3110/8/12/680convex function (CF)schur inequalityHermite–Hadamard inequalities |
| spellingShingle | Ali N. A. Koam Ammara Nosheen Khuram Ali Khan Mudassir Hussain Bukhari Ali Ahmad Maryam Salem Alatawi On Riemann–Liouville Integral via Strongly Modified (<i>h</i>,<i>m</i>)-Convex Functions Fractal and Fractional convex function (CF) schur inequality Hermite–Hadamard inequalities |
| title | On Riemann–Liouville Integral via Strongly Modified (<i>h</i>,<i>m</i>)-Convex Functions |
| title_full | On Riemann–Liouville Integral via Strongly Modified (<i>h</i>,<i>m</i>)-Convex Functions |
| title_fullStr | On Riemann–Liouville Integral via Strongly Modified (<i>h</i>,<i>m</i>)-Convex Functions |
| title_full_unstemmed | On Riemann–Liouville Integral via Strongly Modified (<i>h</i>,<i>m</i>)-Convex Functions |
| title_short | On Riemann–Liouville Integral via Strongly Modified (<i>h</i>,<i>m</i>)-Convex Functions |
| title_sort | on riemann liouville integral via strongly modified i h i i m i convex functions |
| topic | convex function (CF) schur inequality Hermite–Hadamard inequalities |
| url | https://www.mdpi.com/2504-3110/8/12/680 |
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