Multivariate Lyapunov inequalities

Functional Inequalities via Lyapunov Conditions

We review here some recent results by the authors, and various coauthors, on (weak,super) Poincaré inequalities, transportation-information inequalities or logarithmic Sobolev inequality via a quite simple and efficient technique: Lyapunov conditions.

On Multivariate Grüss Inequalities

provided that f and g are two integrable functions on a, b such that p ≤ f x ≤ P , q ≤ g x ≤ Q, for all x ∈ a, b , where p, P, q, Q are real constants. Many generalizations, extensions, and variants of this inequality 1.1 have appeared in the literature, see 1–8 and the references given therein. The main purpose of the present paper is to establish several multivariate Grüss integral inequaliti...

Inequalities for Multivariate Polynomials

We summarize researches – in great deal jointly with my host Y. Sarantopoulos and his PhD students V. Anagnostopoulos and A. Pappas – started by a Marie Curie fellowship in 2001 and is still continuing. The project was to study multivariate polynomial inequalities. In the course of work we realized the role of the “generalized Minkowski functional”, to which we devoted a throughout survey. Resu...

Multivariate Trace Inequalities

Abstract We prove several trace inequalities that extend the Golden-Thompson and the Araki-LiebThirring inequality to arbitrarily many matrices. In particular, we strengthen Lieb’s triple matrix inequality. As an example application of our four matrix extension of the Golden-Thompson inequality, we prove remainder terms for the monotonicity of the quantum relative entropy and strong sub-additiv...

Inertia theorems for operator Lyapunov inequalities