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.