Matrix inequalities from a two variables functional

We introduce a two variables norm functional and establish its joint log-convexity. This entails and improves many remarkable matrix inequalities, most of them related to the log-majorization theorem of Araki. In particular: if[Formula: see text] is a positive semidefinite matrix and[Formula: see text] is a normal matrix,[Formula: see text] and[Formula: see text] is a subunital positive linear map, then[Formula: see text] is weakly log-majorized by[Formula: see text]. This far extension of Araki’s theorem (when [Formula: see text] is the identity and [Formula: see text] is positive) complements some recent results of Hiai and contains several special interesting cases, such as a triangle inequality for normal operators and some extensions of the Golden–Thompson trace inequality. Some applications to Schur products are also obtained.

Recent results on the majorization theory of graph spectrum and topological index theory

Suppose π = (d_1,d_2,...,d_n) and π′ = (d′_1,d′_2,...,d′_n) are two positive non- increasing degree sequences, write π ⊳ π′ if and only if π \neq π′, \sum_{i=1}^n d_i = \sum_{i=1}^n d′_i, and \sum_{i=1}^j d_i ≤ \sum_{i=1}^j d′_i for all j = 1, 2, . . . , n. Let ρ(G) and μ(G) be the spectral radius and signless Laplacian spectral radius of G, respectively. Also let G and G′ be the extremal graphs with the maximal (signless Laplacian) spectral radii in the class of connected graphs with π and π′ as their degree sequences, respectively. If π ⊳ π′ can deduce that ρ(G) < ρ(G′) (respectively, μ(G) < μ(G′)), then it is said that the spectral radii (respectively, signless Laplacian spectral radii) of G and G′ satisfy the majorization theorem. This paper presents a survey to the recent results on the theory and application of the majorization theorem in graph spectrum and topological index theory.

Integral Majorization Theorem for Invex Functions

We obtain some general inequalities and establish integral inequalities of the majorization type for invex functions. We give applications to relative invex functions.