In this paper we extend Jensen-Steffensen’s inequality, via majorization arguments, into the framework of \(\mathbb {R}^n\) , for any \(n\ge 2\) . Generally speaking, our aim is to prove convex type inequalities by relaxing the weights, which are allowed to be nonpositive. We are dealing with monotonic increasing or decreasing sequences with respect to majorization relation in \(\mathbb {R}^n\) and their behaviour under convex functions invariant on permutation of variables. More precisely, Jensen-Steffensen’s and Sherman’s type inequalities are obtained, even in the context of strongly convex functions. Moreover, several further developments concerning extensions on spaces with curved geometry and relative convexity aspects are also discussed.
