Hermite-Hadamard's Type Inequalities for Superquadratic Interval-Valued Functions via Multiplicative Fractional Interval Calculus

This paper considers a new class of superquadraticity called multiplicatively (superquadratic interval-valued functions) superquadratic 𝓘𝓥𝓕s. For these superquadratic 𝓘𝓥𝓕, we prove novel fractional integral inequalities of Hermite-Hadamard (𝓗:𝓗) type using the multiplicative calculus framework. Additionally, using the same calculus framework, we expand our analysis to get fractional inequalities for the product and quotient of multiplicatively superquadratic and subquadratic 𝓘𝓥𝓕s. For multiplicatively superquadratic 𝓘𝓥𝓕, the findings naturally reduce to their corresponding integer-order forms when ρ = 1. We provide numerical calculations and graphical representations based on many illustrative cases to corroborate our theoretical conclusions, demonstrating the solutions' practical applicability and resilience. Furthermore, we investigate possible uses of these inequalities, especially when it comes to linear combinations of special means. This broadens the use of multiplicative convex analysis and offers a new viewpoint on superquadratic 𝓘𝓥𝓕s. Within the context of fractional multiplicative calculus, the conclusions provided in this paper are completely novel and have never been documented in the literature before. We think that our work will open up new avenues for investigation, providing strong tools for mathematical modelling incorporating 𝓘𝓥𝓕s and a greater comprehension of convexity phenomena.