Integral Jensen–Mercer and Related Inequalities for Signed Measures with Refinements

In this paper, we give necessary and sufficient conditions for the integral Jensen–Mercer inequality and closely related inequalities to be satisfied for finite signed measures. As applications, we obtain new special inequalities that are related to the integral Jensen–Steffensen inequality. We also provide refinements of the majorization-type inequality associated with the Jensen–Mercer inequality for finite signed measures. Using the result obtained, we extend a known refinement. The majorization-type inequalities needed for the proofs are interesting in themselves.