Abstract

In this paper, we construct a new parameterized integral kernel based on a proportional Caputo–hybrid–tempered–conformable kernel architecture. The proposed kernel combines proportional Caputo–hybrid weighting, conformable geometric scaling, and exponential tempering within a single kernel design. This formulation provides a flexible setting for fractional integral inequalities and induces an integral operator whose fundamental properties, including linearity, compatibility, and well-definedness, are established. We first derive a fundamental Milne-type identity corresponding to the new kernel. Using this identity, new upper bounds are obtained under the assumption that f′ and f′′, as well as f′q and f′′q, are convex. The resulting inequalities are formulated through the tempered–conformable kernel and provide quantitative estimates for the deviation between fractional integral averages and Milne–Mercer-type expressions. Several limiting cases are also examined. Several existing inequalities involving proportional Caputo–hybrid, tempered, and conformable kernels arise naturally as limiting parameter configurations of the present construction. Consequently, the present results extend existing Milne–Mercer inequalities through a new parameterized kernel design, while preserving several important limiting cases.

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