Inequalities for multiplicative generalized 𝑘-fractional conformable integrals on convex functions

Abstract In this paper, we introduce the multiplicative generalized 𝑘-fractional conformable (MK-FCI) left and right integrals and investigate their fundamental properties, including integrability, boundedness, continuity, and the semigroup property. We then establish Hermite–Hadamard (HH) inequalities for these integrals, and further derive midpoint and trapezoidal inequalities as consequences of the obtained Hermite–Hadamard results. We also show how these new inequalities generalize and connect to existing results for multiplicative generalized 𝑘-fractional conformable integrals and classical multiplicative integrals through particular cases. Furthermore, we present novel Hermite–Hadamard, midpoint, and trapezoidal-type inequalities specifically tailored to the MK-FCI framework.