Hardy–Littlewood–Pólya-type theorems for invex functions

In this paper, the Hardy–Littlewood–Pólya theorem on majorization is extended from convex functions to invex ones. Some variants for pseudo-invex and quasi-invex functions are also considered. The framework used is that of similarly separable vectors. The results obtained are illustrated for monotonic, monotonic in mean, and star-shaped vectors, respectively. Applications to relative invexity are given.