A note on “Concave function inequalities for sums of matrices”

Discussion on: “Linearization Algorithm for a Reduced Order H∞ Control Design of an Active Suspension System”

In this paper, we present some norm inequalities for concave functions, which generalize the main results in [Y. Zhang. Linear Algebra Appl., 574:60-66, 2019].

On some maximal inequalities for demimartingales and [formula omitted]-demimartingales based on concave Young functions

Stability of discrete-time delayed systems via convex function-based summation inequality

In this paper, we present several singular value inequalities for special types of functions of products and sums of matrices. Some of special cases of our results give a generalization of some recent inequalities

New stability analysis for discrete time-delay systems via auxiliary-function-based summation inequalities

The power means are defined using the convex, or concave, power, logarithmic and exponential functions. In this chapter means are defined using arbitrary convex and concave functions by a natural extension of the classical definitions and analogues of the basic results of the earlier chapters are investigated. First however we take up the problem of different convex functions defining the same means; the case of equivalent means. The generalizations (GA) and (r;s), their converses and the Rado-Popoviciu type extensions are studied under the topic of comparable means. The definition can be further extended although this leads to the topics of functional equations and functional inequalities so is not followed in detail.

George Lorentz influenced the author’s research on inequalities in approximation in many ways. This is the connecting thread of this survey paper. The themes of the survey are listed at the very beginning of the paper.

This short but self-contained survey presents a number of elegant matrix/operator inequalities for general convex or concave functions, obtained with a unitary orbit technique. Jensen-, sub- or superadditivity-type inequalities are considered. Some of them are substitutes to classical inequalities (Choi, Davis, Hansen–Pedersen) for operator convex or concave functions. Various trace, norm and determinantal inequalities are derived. Combined with an interesting decomposition for positive semi-definite matrices, several results for partitioned matrices are also obtained.

Some inequalities for sum and product of positive semidefinite matrices

In this paper, we introduce several numerical radius inequalities involving off-diagonal part of 2 ? 2 positive semidefinite block matrices and their diagonal blocks. It is shown that if A, B,C ? Mn(C) are such that [A B* B C] ? 0, then w2r(B) ? 1/2 ?||A4r? + A4r(1??)|| ||C4r? + C4r(1??)|| , and w2r(B) ? ||?A r/? + (1 ? ?)C r/1?? , for 0 < ? < 1, r ? 1. Moreover, we establish some numerical radius inequalities for products and sums of matrices.

This paper establishes new singular value inequalities for products and sums of matrices. We consider a class of matrices in and present several inequalities that generalize and refine existing results in the literature. The presented inequalities offer a sharper perspective on the relationships between the singular values of these matrices.

Singular value and norm inequalities for products and sums of matrices

For compact Hilbert space operators A and B, the singular values of $A^ * B$ are shown to be dominated by those of $\frac{1}{2}(AA^* + BB^* )$.

Sub-additive and super-additive inequalities for concave and convex functions have been generalized to the case of matrices by several authors over a period of time. These lead to some interesting inequalities for matrices, which in some cases coincide with, and in other cases are at variance with the corresponding inequalities for real numbers. We survey some of these matrix inequalities and do further investigations into these. We introduce the novel notion of dominated majorization between the spectra of two Hermitian matrices B and C, dominated by a third Hermitian matrix A. Based on an explicit formula for the gradient of the sum of the k largest eigenvalues of a Hermitian matrix, we show that under certain conditions dominated majorization reduces to a linear majorization-like relation between the diagonal elements of B and C in a certain basis. We use this notion as a tool to give new, elementary proofs for the sub-additivity inequality for non-negative concave functions first proved by Bourin and Uchiyama and the corresponding super-additivity inequality for non-negative convex functions first proven by Kosem. Finally, we present counterexamples to some conjectures that Ando's inequality for operator convex functions could more generally hold, e.g. for ordinary convex, non-negative functions.