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></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.