Multiplicative Fractional Milne-Mercer-Type Inequalities via Multiplicative Atangana-Baleanu-Conformable Integral Operators

This paper introduces a multiplicative Atangana–Baleanu–conformable fractional integral operator in the setting of multiplicative calculus. The proposed operator is formulated by applying the Atangana–Baleanu–conformable fractional integral structure to the logarithmic representation of positive functions, thereby combining multiplicative behavior, nonsingular memory effects, and conformable scaling in a single framework. Appropriate function-space assumptions are imposed to ensure that the operator is well defined. Based on this operator, we establish a new auxiliary identity and derive several multiplicative Milne–Mercer-type inequalities for multiplicatively convex functions. The obtained results include multiplicative Riemann–Liouville-type, multiplicative Atangana–Baleanu-type, and conformable-type inequalities as special cases under suitable choices of the parameters. To clarify the role of the fractional parameters, numerical examples are provided together with logarithmic gap values, relative-error comparisons, heatmaps, contour plots, and parameter-sensitivity analyses. These computations illustrate the validity of the derived inequalities and compare the proposed bounds with their reduced special cases.