| Summary: | Convexity is crucial in obtaining many forms of inequalities. As a result, there is a significant link between convexity and integral inequality. Due to the significance of these concepts, the purpose of this study is to introduce a new class of generalized convex interval-valued functions called <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo>(</mo><mrow><mi>p</mi><mo>,</mo><mi>s</mi></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>-convex fuzzy interval-valued functions (<inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo>(</mo><mrow><mi>p</mi><mo>,</mo><mi>s</mi></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>-convex <i>F-I-V-F</i>s) in the second sense and to establish Hermite–Hadamard (H–H) type inequalities for <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo>(</mo><mrow><mi>p</mi><mo>,</mo><mi>s</mi></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>-convex <i>F-I-V-F</i>s using fuzzy order relation. In addition, we demonstrate that our results include a large class of new and known inequalities for <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo>(</mo><mrow><mi>p</mi><mo>,</mo><mi>s</mi></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>-convex <i>F-I-V-F</i>s and their variant forms as special instances. Furthermore, we give useful examples that demonstrate usefulness of the theory produced in this study. These findings and diverse approaches may pave the way for future research in fuzzy optimization, modeling, and interval-valued functions.
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