ANALYSIS ON MULTIPLICATIVE k-ATANGANA–BALEANU FRACTIONAL INTEGRALS WITH APPLICATION TO VARIOUS MERCER-TYPE INEQUALITIES

We innovatively introduce a new group of integral operators called multiplicative [Formula: see text]-Atangana–Baleanu fractional integrals. Following the path of this research, we discuss their [Formula: see text]integrability, continuity, boundedness together with linearity, and successfully construct two Hermite–Hadamard–Mercer inequalities employing this multiplicative fractional integrals, focusing on endpoint- and midpoint-type, respectively. Subsequently, we further propose three identities, involving trapezoid-Mercer-, midpoint-Mercer-, and parameterized-Mercer-type, and combining the function [Formula: see text] is multiplicatively convex or, for some fixed [Formula: see text], the function [Formula: see text] is convex, and we formulate a series of corresponding fractional Mercer-type inequalities. Intended to enhance the readers’ deeper understanding of the results, we offer two instances accompanied by graphical representations, facilitating the validity of the inequalities gained in the current paper. Finally, some applications in multiplicative differential equation, the quadrature formula and special means are presented as well.