Refinements of a reversed AM-GM operator inequality

Banach Journal of Mathematical Analysis, 2013

In this paper we derive some improvements of means inequalities for Hilbert space operators. More precisely, we obtain refinements and reverses of the arithmetic-geometric operator mean inequality. As an application, we also deduce an improved variant for the refined arithmetic-Heinz mean inequality. We also present some eigenvalue inequalities for differences of certain operator means.

Linear Algebra and its Applications, 2006

Two families of means (called Heinz means and Heron means) that interpolate between the geometric and the arithmetic mean are considered. Comparison inequalities between them are established. Operator versions of these inequalities are obtained. Failure of such extensions in some cases is illustrated by a simple example.

2014

In this paper, we provide some interested operator inequalities related with non-negative linear maps by means of concavity and convexity structure, and also establish some new attractive inequalities for the Khatri-Rao products of two or more positive definite matrices. These results lead to inequalities for Hadamard product and Ando's and a-power geometric means, as a special case.

Banach Journal of Mathematical Analysis, 2015

In this paper some inequalities involving quasi-arithmetic means for a continuous field of self-adjoint operators, a field of positive linear mappings and continuous strictly monotone functions are refined. These refined converses are presented by using the Mond-Pečarić method improvement. Obtained results are applied to refine selected inequalities with power functions.

arXiv: Functional Analysis, 2020

In this survey, we shall present characterizations of some distinguished classes of Hilbertian bounded linear operators (namely, normal operators, selfadjoint operators, and unitary operators) in terms of operator inequalities related to the arithmetic-geometric mean inequality. For the class of all normal operators, we shall present new general characterizations.

Linear Algebra and its Applications, 1994

Porta, and Recht recently proved that (ISTS-' + S-'TSIj > 21jTI(. A generalization of this inequality to larger classes of operators and norms is obtained as an immediate consequence of the operator form of the arithmetic-geometric-mean inequality. Some related inequalities are also discussed. 1.

Acta Mathematica Sinica, English Series, 2017

Following an idea of Lin, we prove that if A and B be two positive operators such that 0 < mI ≤ A ≤ m ′ I ≤ M ′ I ≤ B ≤ M I, then 2010 Mathematics Subject Classification. 47A63, 47A30, 47A64, 15A63.

Journal of inequalities and applications, 2018

In this paper, we study some complementary inequalities to Jensen&#39;s inequality for self-adjoint operators, unital positive linear mappings, and real valued twice differentiable functions. New improved complementary inequalities are presented by using an improvement of the Mond-Pečarić method. These results are applied to obtain some inequalities with quasi-arithmetic means.

In this note, we derive non trivial sharp bounds related to the weighted harmonic-geometric-arithmetic means inequalities, when two out of the three terms are known. As application, we give an explicit bound for the trace of the inverse of a symmetric positive definite matrix and an inequality related to the coefficients of polynomials with positive roots.

arXiv (Cornell University), 2022

Let A i and B i be positive definite matrices for all i = 1, • • • , m. It is shown that for all unitarly invariant norms, for all p ≥ 1 and for all r ≥ 1, where A, B, C, D are positive definite matrices. This gives an affirmative answer to the conjecture posed by Dinh, Ahsani and Tam in the case of m = 2. The preceding inequalities directly lead to a recent result of Audenaert [3].

Linear Algebra and Its Applications, 2006

A variant of Jensen’s operator inequality for convex functions, which is a generalization of Mercer’s result, is proved. Obtained result is used to prove a monotonicity property for Mercer’s power means for operators, and a comparison theorem for quasi-arithmetic means for operators.

Filomat, 2012

In this paper we study inequalities among quasi-arithmetic means for a continuous field of self-adjoint operators, a field of positive linear mappings and continuous strictly monotone functions which induce means. We present inequalities with operator convexity and without operator convexity of appropriate functions. Also, we present a general formulation of converse inequalities in each of these cases. Furthermore, we obtain refined inequalities without operator convexity. As applications, we obtain inequalities among power means.

Scientiae Mathematicae Japonicae, 2010

As a continuation of our previous research [Sci. Math. Japon. 61 (2005), 25-46.], we discuss order among power means of positive operators with a unital ntuple of positive linear maps. 1 Introduction. We assume that H and K are Hilbert spaces and B(H) and B(K) are C*-algebras of all bounded linear operators on the appropriate Hilbert space. We say that an n-tuple (Φ 1 ,. .. , Φ n) of positive linear maps Φ i : B(H) → B(K) is unital if n i=1 Φ i (1) = 1. Recently F. Hansen, J. Pečarić and I. Perić in [4, 5] gave a general formulation of Jensen's operator inequality for unital fields of positive linear mappings and its converses. Their main results from [4] are presented in the following two theorems [4, Theorem 1 and Theorem 2 ]. 2000 Mathematics Subject Classification. 47A64, 47A63. Key words and phrases. generalization of Jensen's operator inequality, generalization of converse of Jensen's operator inequality, operator order, power operator means.

We establish matrix versions of refinements due to Alzer [1], Cartwright and Field [4], and Mercer [5] of the standard arithmetic-geometric-harmonic mean inequality for scalars.