Milne's Estimates via Conformable Fractional Multiplicative Integrals

This paper explores Milne’s inequality in the context of multiplicative conformable fractional integrals, a recent extension of classical fractional calculus rooted in non-Newtonian analysis. Leveraging recent developments in multiplicative calculus, we establish a new fundamental identity that underpins the derivation of Milne-type inequalities for two classes of functions: (i) those whose $^*$-derivatives in $^*$-absolute value are multiplicative convex, and (ii) those with bounded $^*$-derivatives. A numerical example, accompanied by graphical illustrations, is included to demonstrate the validity and effectiveness of the theoretical findings. Further, we extend our analysis to functions satisfying specific derivative constraints and present some applications to special means. The paper concludes with a summary of the main contributions and a discussion of promising avenues for future research in multiplicative fractional analysis.