Fuzzy Convexity Under <i>cr</i>-Order with Control Operator and Fractional Inequalities
This study is organized to introduce the concept of center–radius <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>c</mi><mi>r</mi><mo>)</mo><...
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2025-06-01
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| author | Qi Liu Muhammad Zakria Javed Muhammad Uzair Awan Loredana Ciurdariu Badr S. Alkahtani |
| author_facet | Qi Liu Muhammad Zakria Javed Muhammad Uzair Awan Loredana Ciurdariu Badr S. Alkahtani |
| author_sort | Qi Liu |
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| description | This study is organized to introduce the concept of center–radius <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>c</mi><mi>r</mi><mo>)</mo></mrow></semantics></math></inline-formula>-ordered fuzzy number-valued convex mappings. Based on this class of mappings, we have initiated the idea of fuzzy number-valued extended <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mi>r</mi></mrow></semantics></math></inline-formula>-<i>ℏ</i> convex mappings incorporating control mapping <i>ℏ</i>. Furthermore, several potential new classes of convexity will be provided to discuss its generic nature. Also, some essential properties, criteria, and detailed characterizations through Jensen’s and Hermite–Hadamard-like inequalities are provided, incorporating Riemann–Liouville fractional operators, which are defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-level mappings. To validate the proposed fractional bounds through simulations, we consider both triangular and trapezoidal fuzzy numbers. Our results are based on totally ordered fuzzy-valued mappings, which are new and generic. The under-consideration class also includes a blend of new classes of convexity, which are controlled by non-negative mapping <i>ℏ</i>. In previous studies, the researchers have focused on different partially ordered relations. |
| format | Article |
| id | doaj-art-05abf2e527034567989b0061308b12c4 |
| institution | Kabale University |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj-art-05abf2e527034567989b0061308b12c42025-08-20T03:24:36ZengMDPI AGFractal and Fractional2504-31102025-06-019639110.3390/fractalfract9060391Fuzzy Convexity Under <i>cr</i>-Order with Control Operator and Fractional InequalitiesQi Liu0Muhammad Zakria Javed1Muhammad Uzair Awan2Loredana Ciurdariu3Badr S. Alkahtani4School of Mathematics and Physics, Anqing Normal University, Anqing 246133, ChinaDepartment of Mathematics, Government College University Faisalabad, Faisalabad 38000, PakistanDepartment of Mathematics, Government College University Faisalabad, Faisalabad 38000, PakistanDepartment of Mathematics, Politehnica University of Timisoara, 300006 Timisoara, RomaniaDepartment of Mathematics, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi ArabiaThis study is organized to introduce the concept of center–radius <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>c</mi><mi>r</mi><mo>)</mo></mrow></semantics></math></inline-formula>-ordered fuzzy number-valued convex mappings. Based on this class of mappings, we have initiated the idea of fuzzy number-valued extended <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mi>r</mi></mrow></semantics></math></inline-formula>-<i>ℏ</i> convex mappings incorporating control mapping <i>ℏ</i>. Furthermore, several potential new classes of convexity will be provided to discuss its generic nature. Also, some essential properties, criteria, and detailed characterizations through Jensen’s and Hermite–Hadamard-like inequalities are provided, incorporating Riemann–Liouville fractional operators, which are defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-level mappings. To validate the proposed fractional bounds through simulations, we consider both triangular and trapezoidal fuzzy numbers. Our results are based on totally ordered fuzzy-valued mappings, which are new and generic. The under-consideration class also includes a blend of new classes of convexity, which are controlled by non-negative mapping <i>ℏ</i>. In previous studies, the researchers have focused on different partially ordered relations.https://www.mdpi.com/2504-3110/9/6/391fuzzy numberconvex mappingcenter–radiusfractional calculusJensen’s inequalityHermite–Hadamard’s inequality |
| spellingShingle | Qi Liu Muhammad Zakria Javed Muhammad Uzair Awan Loredana Ciurdariu Badr S. Alkahtani Fuzzy Convexity Under <i>cr</i>-Order with Control Operator and Fractional Inequalities Fractal and Fractional fuzzy number convex mapping center–radius fractional calculus Jensen’s inequality Hermite–Hadamard’s inequality |
| title | Fuzzy Convexity Under <i>cr</i>-Order with Control Operator and Fractional Inequalities |
| title_full | Fuzzy Convexity Under <i>cr</i>-Order with Control Operator and Fractional Inequalities |
| title_fullStr | Fuzzy Convexity Under <i>cr</i>-Order with Control Operator and Fractional Inequalities |
| title_full_unstemmed | Fuzzy Convexity Under <i>cr</i>-Order with Control Operator and Fractional Inequalities |
| title_short | Fuzzy Convexity Under <i>cr</i>-Order with Control Operator and Fractional Inequalities |
| title_sort | fuzzy convexity under i cr i order with control operator and fractional inequalities |
| topic | fuzzy number convex mapping center–radius fractional calculus Jensen’s inequality Hermite–Hadamard’s inequality |
| url | https://www.mdpi.com/2504-3110/9/6/391 |
| work_keys_str_mv | AT qiliu fuzzyconvexityundericriorderwithcontroloperatorandfractionalinequalities AT muhammadzakriajaved fuzzyconvexityundericriorderwithcontroloperatorandfractionalinequalities AT muhammaduzairawan fuzzyconvexityundericriorderwithcontroloperatorandfractionalinequalities AT loredanaciurdariu fuzzyconvexityundericriorderwithcontroloperatorandfractionalinequalities AT badrsalkahtani fuzzyconvexityundericriorderwithcontroloperatorandfractionalinequalities |