On Milne–Mercer-Type Inequalities Associated with Atangana–Baleanu-Tempered–Conformable Integral Operators

In this paper, we introduce Atangana–Baleanu–tempered–conformable (ABTC) generalized integral operators and establish associated Milne–Mercer-type inequalities for differentiable convex functions. The proposed kernel simultaneously incorporates AB-type local–nonlocal mixing, exponential attenuation, and conformable distance scaling. Using a kernel-dependent integral identity together with the Jensen–Mercer inequality, convexity arguments, and Hölder’s inequality, we derive estimates whose coefficients depend jointly on the independent parameters θ, α, β, and λ. None of the corresponding AB–conformable, tempered–conformable, or AB-type tempered specializations retains all these mechanisms simultaneously. Parameter reductions recover several previously known integral settings and provide consistency checks for the construction. Numerical examples are included to illustrate the validity, comparative behavior, and parameter sensitivity of the resulting bounds.