Further refinements of the Heinz inequality

“…For example, in [8], the convexity of the function t → ||| ≤ |||A t XB 1−t + A 1−t XB t ||| was utilized to find some refining terms of (1.2). Then in [4,12], further refinements were obtained, modeling the same idea of [8]; see also [1,14]. For example, it was shown in [8] that…”

Section: Introductionmentioning

confidence: 99%

Quadratic interpolation of the Heinz means

Kittaneh¹,

Moslehian²,

Sababheh³

2018

Mathematical Inequalities &Amp; Applications

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The main goal of this article is to present several quadratic refinements and reverses of the well known Heinz inequality, for numbers and matrices, where the refining term is a quadratic function in the mean parameters. The proposed idea introduces a new approach to these inequalities, where polynomial interpolation of the Heinz function plays a major role. As a consequence, we obtain a new proof of the celebrated Heron-Heinz inequality proved by Bhatia, then we study an optimization problem to find the best possible refinement. As applications, we present matrix versions including unitarily invariant norms, trace and determinant versions.2010 Mathematics Subject Classification. 15A39, 15B48, 47A30, 47A63.

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“…known as the Heinz mean inequality; see [1,4] and references therein for more information on Heinz inequality. The organization of the paper is as follows.…”

Section: Introductionmentioning

confidence: 99%

Advanced refinements of Young and Heinz inequalities

Sababheh

Moslehian

2017

Journal of Number Theory

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In this article, we prove several multi-term refinements of Young type inequalities for both real numbers and operators improving several known results. Among other results, we prove that for all 0 ≤ ν ≤ 1 and each N ∈ N,for the positive operators A and B, where s j (ν) and α j (ν) are certain functions. Moreover, some new Heinz type inequalities involving the Hilbert-Schmidt norm are established.

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Further refinements of the Heinz inequality

Cited by 31 publications

References 20 publications

Some generalized numerical radius inequalities for Hilbert space operators

Some generalized numerical radius inequalities for Hilbert space operators

Quadratic interpolation of the Heinz means

Advanced refinements of Young and Heinz inequalities

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