Jensen’s inequality for operators without operator convexity

Mičić, Jadranka; Pavić, Zlatko; Pečarić, Josip

doi:10.1016/j.laa.2010.11.004

Cited by 22 publications

(15 citation statements)

References 4 publications

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“…When p = 0, this means exp(Φ 1 (log A)+Φ 2 (log B)) by (2.2). Therefore, the next result is a special case of Theorem 2.1, which was shown in [12,13,14] (see also [6,Chapter 4]). In fact, results in more general forms were given there.…”

Section: Results and Motivationmentioning

confidence: 67%

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On matrix inequalities between the power means: Counterexamples

Audenaert

Hiai

2013

Linear Algebra and its Applications

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We prove that the known sufficient conditions on the real parameters (p, q) for which the matrix power mean inequality ((A p + B p )/2) 1/p ≤ ((A q + B q )/2) 1/q holds for every pair of matrices A, B > 0 are indeed best possible. The proof proceeds by constructing 2 × 2 counterexamples. The best possible conditions on (p, q) for which Φ(A p ) 1/p ≤ Φ(A q ) 1/q holds for every unital positive linear map Φ and A > 0 are also clarified.2010 Mathematics Subject Classification: Primary 15A45, 47A64

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Section: Results and Motivationmentioning

confidence: 67%

“…A more general result involving positive linear maps is known under suitable assumptions on p, q in [6,12,13,14] (see Theorems 2.1 and 2.2 below). Our interest here is showing that these sufficient conditions of p, q are best possible for (1.3) to hold.…”

Section: Introductionmentioning

confidence: 99%

On matrix inequalities between the power means: Counterexamples

Audenaert

Hiai

2013

Linear Algebra and its Applications

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“…Proof. We use an idea from [11,Theorem 1]. For simplicity, let a " n∇ v m and b " N∇ v M. Now since ra, bs X rm, Ms " ∅ and ra, bs X rn, Ns " ∅, we will consider the secant of f on the interval ra, bs.…”

Section: Resultsmentioning

confidence: 99%

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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“…Quasi-arithmetic operator means without applying operator convexity were also investigated in [4,10].…”

Section: Application To Quasi-arithmetic Meansmentioning

confidence: 99%

“…The operator inequality of (2) was formulated for convex (without operator) continuous functions in [4] assuming the spectral conditions: Sp( ) ⊆ [ , ] and Sp( ) ∩ ( , ) = 0 for all , where = ∑ =1 Φ ( ).…”

Section: Introductionmentioning

confidence: 99%