On tempered fractional Bullen-type inequalities: analysis in different function settings

Tempered fractional integrals offer a flexible and physically realistic extension of classical fractional operators by incorporating an exponential tempering factor that attenuates long-range memory effects. This modification not only ensures better convergence and numerical stability but also allows for more accurate modeling of real-world phenomena such as anomalous diffusion, financial returns, or geophysical processes, where power-law behaviors are observed but truncated at large scales. This study advances Bullen-type inequalities within the framework of tempered fractional calculus. By formulating a novel identity based on tempered operators, we derive generalized estimates applicable to various function classes, including those with convex or bounded derivatives, r-L-Hölderian functions, and functions of bounded variation. The inclusion of exponential tempering parameters refines classical Bullen results by balancing non-local effects with convergence guarantees. To illustrate their applicability, these bounds are applied to special means, highlighting how convexity constraints help stabilize tempered fractional integrals in practical contexts.