Some Norm Inequalities for Completely Monotone Functions

Aujla, Jaspal Singh

doi:10.1137/s0895479800369761

Cited by 7 publications

(7 citation statements)

References 8 publications

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“…Inspired by the result given in Theorem 2.1, we present some extensions of the inequalities due to Aujla and Silva [4]. Our inequalities refine earlier results in this direction obtained in [3] and [10].…”

Section: Introductionsupporting

confidence: 76%

“…The following corollary presents a stronger version of [3,Corollary 2.5]. In [3], unitarily invariant versions for complex matrices have been obtained.…”

Section: Resultsmentioning

confidence: 97%

“…Another result of this type is due to Aujla [3] which asserts that if the function f is completely monotone (in the sense, p´1q k f pkq pxq ě 0 for all k " 0, 1, . .…”

Section: Introductionmentioning

confidence: 99%

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Subadditive inequalities for operators

Moradi¹,

Heydarbeygi²,

Sababheh³

2020

Mathematical Inequalities &Amp; Applications

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In this article, we present a new subadditivity behavior of convex and concave functions, when applied to Hilbert space operators. For example, under suitable assumptions on the spectrum of the positive operators A and B, we prove that 2 1´r pA`Bq r ď A r`Br for r ą 1 and r ă 0, and A r`Br ď 2 1´r pA`Bq r for r P r0, 1s .These results provide considerable generalization of earlier results by Aujla and Silva.Further, we present several extensions of the subadditivity idea initiated by Ando and Zhan then extended by Bourin and Uchiyama.A real-valued continuous function f on an interval J is said to be operator convex (resp.operator concave) if f pA∇ v Bq ď presp. ěq f pAq ∇ v f pBq for all v P r0, 1s and for all selfadjoint operators A, B P BpHq whose spectra are contained in J. A continuous function f on J is called operator monotone increasing (resp. decreasing), ifA ď B ñ f pAq ď presp. ěq f pBq .

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Section: Introductionsupporting

confidence: 76%

“…The following corollary presents a stronger version of [3,Corollary 2.5]. In [3], unitarily invariant versions for complex matrices have been obtained.…”

Section: Resultsmentioning

confidence: 97%

See 1 more Smart Citation

Subadditive inequalities for operators

Moradi¹,

Heydarbeygi²,

Sababheh³

2020

Mathematical Inequalities &Amp; Applications

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show abstract

“…It is shown in [3,5] that if a non-negative function f satisfies (1) then it is concave and if it satisfies (2) then it is convex with f (0) = 0. Hence within the set of non-negative f these results give a full characterisation of all possible f satisfying these inequalities.…”

Section: Comparison Of Norms |||Fmentioning

confidence: 99%

On norm sub-additivity and super-additivity inequalities for concave and convex functions

Audenaert

Aujla

2012

Linear and Multilinear Algebra

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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 nonnegative 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.Dedicated to the memory of Ky Fan

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“…We emphasize that (1.6) does not hold without the norm. In [4], it is shown that an operator monotone decreasing function f : (0, ∞) → (0, ∞) satisfies the subadditive inequality…”

Section: Introductionmentioning

confidence: 99%

Some inequalities for interpolational operator means

Moradi,

Furuichi,

Sababheh

2018

Preprint

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Using the properties of geometric mean, we shall show for any 0 ≤ α, β ≤ 1,whenever f is a non-negative operator log-convex function, A, B ∈ B (H) are positive operators, and 0 ≤ α, β ≤ 1. As an application of this operator mean inequality, we present several refinements of the Aujla subadditive inequality for operator monotone decreasing functions. Also, in a similar way, we consider some inequalities of Ando's type. Among other things, it is shown that if Φ is a positive linear map, then

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Some Norm Inequalities for Completely Monotone Functions

Cited by 7 publications

References 8 publications

Subadditive inequalities for operators

Subadditive inequalities for operators

On norm sub-additivity and super-additivity inequalities for concave and convex functions

Some inequalities for interpolational operator means

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