Some improvements on the Ky Fan theorem for tensors

The improvements of Ky Fan theorem are given for tensors. First, based on Brauer-type eigenvalue inclusion sets, we obtain some new Ky Fan-type theorems for tensors. Second, by characterizing the ratio of the smallest and largest values of a Perron vector, we improve the existing results. Third, some new eigenvalue localization sets for tensors are given and proved to be tighter than those presented by Li and Ng (Numer Math 130(2):315–335, 2015) and Wang et al. (Linear Multilinear Algebra 68(9):1817–1834, 2020). Finally, numerical examples are given to validate the efficiency of our new bounds.

Inequalities for certain Classes of Convex Functions

Making use of properties of doubly-stochastic matrices, I recently gave a simple proof (4) of a theorem of Ky Fan (Theorem 2b below) on symmetric gauge functions. I now propose to show that the same idea can be employed to derive a whole series of results on convex functions ; in particular, certain well-known inequalities of Hardy-Littlewood-Pólya and of Pólya will emerge as specìal cases.