Fractional integral inequalities for strongly convex functions via multiplicative calculus with applications

In this paper, we introduce the concept of strongly multiplicative convex functions and establish Hermite–Hadamard (HH)-type integral inequalities using the Atangana–Baleanu (AB) fractional integral operator within the framework of multiplicative calculus. We further derive HH-type integral inequalities for the product and quotient of strongly multiplicative convex and strongly multiplicative concave functions using the same operator. By setting $\alpha =1$ , we obtain simplified versions of these inequalities for the product and quotient of such functions, which represent novel results not previously explored. The validity of the proposed theorems is demonstrated graphically through well-chosen examples. Additionally, we present a proof of Milne’s inequality utilizing the same operator within the context of multiplicative calculus. To further enhance the study, we explore applications of these functions in relation to special functions and derive new fractional recurrence relations, adding depth and significance to the findings.