Weighted arithmetic–geometric operator mean inequalities

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R E S E A R C H

Open Access

Weighted arithmetic–geometric operator

mean inequalities

Jianming Xue

1*

*_{Correspondence:}

[email protected]

1_{Oxbridge College, Kunming}

University of Science and Technology, Kunming, P.R. China

Abstract

In this paper, we reﬁne and generalize some weighted arithmetic–geometric operator mean inequalities due to Lin (Stud. Math. 215:187–194,2013) and Zhang (Banach J. Math. Anal. 9:166–172,2015) as follows: LetAandBbe positive operators. If 0 <m≤A≤m<M≤B≤Mor 0 <m≤B≤m<M≤A≤M, then for a positive unital linear map

,

2₍_A_∇

αB)≤

_K₍_h₎

S(hr₎

2

2₍_A

αB),

2₍_A_∇

αB)≤

_K₍_h₎

S(hr₎

2

(A)

α

(B)

2

,

2p(A∇αB)≤ 1 16

_K2₍_h₎₍_M2₊_m2₎2

S2₍_hr₎_M2_m2

p

2p(A

αB),

2p₍_A_∇

αB)≤ 1 16

_K2₍_h₎₍_M2₊_m2₎2

S2₍_hr₎_M2_m2

p

(A)

α

(B)

2p

,

where

α

∈[0, 1],K(h) =(h+1)_4h2,S(h) = h 1 h–1

elogh

1 h–1

,h=M_m,h=_mM,r= min{

α

, 1 –

α

}and

p≥2.

MSC: 47A63; 47A30

Keywords: Positive linear map; Operator inequality; Weighted arithmetic operator mean; Weighted geometric operator mean

1 Introduction

LetB(H) be theC∗-algebra of all bounded linear operators on a Hilbert space (H,·,· ) andI be the identity operator. · is the operator norm.A≥0 (A> 0) implies thatA

is a positive (strictly positive) operator. A linear map:B(H)→B(K) is called positive ifA≥0 implies(A)≥0. It is said to be unital if(I) =I. ForA,B> 0, theα-weighted arithmetic mean andα-weighted geometric mean ofAandBare deﬁned, respectively, by

A∇αB= (1 –α)A+αB, AαB=A

1

2_A–12_BA–12α_A12_,

whereα∈[0, 1]. When α= 1₂, we writeA∇BandABfor brevity forA∇1

2BandA12B, respectively.

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Let 0 <m≤A,B≤M, andbe a positive unital linear map. Tominaga [3] showed that the following operator inequality holds:

A+B

2 ≤S(h)AB, (1.1)

whereS(h) = h 1 h–1

eloghh–11

is called Specht’s radio andh=M_m. Indeed

S(h)≤K(h) =(h+ 1)

2

4h ≤S

2₍_h₎ ₍_h_≥₁₎ _(1.2)

was observed by Lin [1, (3.3)].

By (1.1) and (1.2), it is easy to obtain the following inequality:

A+B

2 ≤K(h)(AB). (1.3)

Lin [1, Theorem 2.1] proved that (1.3) can be squared as follows:

2

A+B

2 ≤K

2₍_h₎2₍_A_B₎ _(1.4)

and

2

A+B

2 ≤K

2₍_h₎₍_A₎₍_B₎2_. _(1.5)

Zhang [2] generalized (1.4) and (1.5) whenp≥2

2p

A+B

2 ≤

[K(h)(M2₊_m2_)]2p 16M2p_m2p

2p₍_A_B₎ _(1.6)

and

2p

A+B

2 ≤

[K(h)(M2₊_m2_)]2p 16M2p_m2p

(A)(B)2p. (1.7)

A great number of results on operator inequalities have been given in the literature, for example, see [4–6] and the references therein.

In this paper, motivated by the aforementioned discussion, we extend (1.4)–(1.7) to the weighted arithmetic–geometric mean. In order to prove our results, we show a new opera-tor weighted arithmetic–geometric mean inequality. Manipulating this operaopera-tor inequal-ity enables us to reﬁne and generalize (1.4)–(1.7). Furthermore, a numerical example is given to demonstrate the eﬀectiveness of the theoretical results.

2 Main results

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Lemma 1([7]) Let A,B> 0.Then the following norm inequality holds:

AB ≤ 1

4A+B

2_. _(2.1)

Lemma 2([8]) Let A> 0.Then for every positive unital linear map,

A–1≥–1(A). (2.2)

Lemma 3([9]) Let A,B> 0.Then for1≤r<∞,

Ar+Br≤(A+B)r. (2.3)

Lemma 4([10]) Let0 <m≤A≤m<M≤B≤M or0 <m≤B≤m<M≤A≤M.

Then for eachα∈[0, 1],

A∇αB≥S

hrAαB, (2.4)

where S(h) = h 1 h–1

elogh 1 h–1

,h=M_m and r=min{α, 1 –α}.

Theorem 1 Let0 <m≤A≤m<M≤B≤M or0 <m≤B≤m<M≤A≤M.Then for eachα∈[0, 1],

A∇αB+MmS

hr(AαB)–1≤M+m, (2.5)

where S(h) = h 1 h–1

elogh 1 h–1

,h=M_m and r=min{α, 1 –α}.

Proof Since

0 <m≤A≤M,

then

(1 –α)(M–A)(m–A)A–1≤0.

That is,

(1 –α)A+MmA–1≤(1 –α)(M+m). (2.6)

Similarly, we get

αB+MmB–1≤α(M+m). (2.7)

Summing up inequalities (2.6) and (2.7), we get

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By (AαB)–1=A–1αB–1and (2.4), we have

A∇αB+MmS

hr(AαB)–1≤A∇αB+MmA–1∇αB–1

≤M+m.

This completes the proof.

Theorem 2 Letbe a positive unital linear map and let A and B be positive operators.If

0 <m≤A≤m<M≤B≤M or0 <m≤B≤m<M≤A≤M,then for eachα∈[0, 1],

2(A∇αB)≤

K(h)

S(hr₎

2

2(AαB), (2.8)

2(A∇αB)≤

K(h)

S(hr₎

2

(A)α(B)

2

, (2.9)

where K(h) =(h₄+1)_h2,S(h) = h 1 h–1

elogh 1 h–1

,h=M_m,h=M_m and r=min{α, 1 –α}.

Proof Inequality (2.8) is equivalent to

(A∇αB)–1(AαB)≤

K(h)

S(hr₎.

By (2.1), (2.2) and (2.5), we have

(A∇αB)MmS

hr–1(AαB)≤

1

4(A∇αB) +MmS

hr–1(AαB)2

≤1

4(A∇αB) +MmS

hr(AαB)–1

2

=1

4

(A∇αB) +MmS

hr(AαB)–1

2

≤1

4(M+m)

2

=1

4(M+m)

2_.

That is,

(A∇αB)–1(AαB)≤

(M+m)2

4MmS(hr₎=

K(h)

S(hr₎.

Thus, (2.8) holds.

Inequality (2.9) is equivalent to

(A∇αB)

(A)α(B)

–1

≤ K(h)

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By (2.1) and (2.5), we have

(A∇αB)MmS

hr₍_A₎ α(B)

–1

≤1

4(A∇αB) +MmS

hr(A)α(B)

–12

=1

4(A)∇α(B) +MmS

hr(A)α(B)

–12

≤1

4(M+m)

2_.

That is,

(A∇αB)

(A)α(B)

–1

≤ K(h)

S(hr₎.

Thus, (2.9) holds.

Theorem 3 Letbe a positive unital linear map and let A and B be positive operators.If

0 <m≤A≤m<M≤B≤M or0 <m≤B≤m<M≤A≤M and2≤p<∞,then for eachα∈[0, 1],

2p(A∇αB)≤

1 16

K2₍_h₎₍_M2₊_m2₎2 S2₍_hr₎_M2_m2

p

2p(AαB), (2.10)

2p(A∇αB)≤

1 16

K2(h)(M2+m2)2

S2₍_hr₎_M2_m2

p

(A)α(B)

2p

, (2.11)

where K(h) =(h₄+1)_h2,S(h) = h 1 h–1

elogh 1 h–1

,h=M_m,h=M_m and r=min{α, 1 –α}.

Proof By (2.8), we have

–2(AαB)≤L2–2(A∇αB), (2.12)

whereL=_SK₍_h(hr)₎.

Inequality (2.10) is equivalent to

p(A∇αB)–p(AαB)≤

1 4

K2₍_h₎₍_M2₊_m2₎2 S2₍_hr₎_M2_m2

p 2 .

By (2.1), (2.3) and (2.12), we have

p(A∇αB)Mpmp–p(AαB)

≤1

4

Lp2p₍_A∇_α_B_{) +}

M2m2 L

p 2

–p(AαB)

2

≤1

4

L2(A∇αB) +

M2_m2

L

–2₍_A

αB)

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≤1 4L

2₍_A_∇

αB) +LM2m2–2(A∇αB) p

≤1

4

LM2+m2p.

That is,

p(A∇αB)–p(AαB)≤

1 4

L(M2₊_m2₎ Mm

p =1

4

K2₍_h₎₍_M2₊_m2₎2 S2₍_hr₎_M2_m2

p 2 .

Thus, (2.10) holds.

Similarly, (2.11) holds by inequality (2.9).

Remark1 Whenα=1₂, because of K(h)

S(√h) <K(h), inequalities (2.8), (2.9), (2.10) and (2.11)

are sharper than (1.4), (1.5), (1.6) and (1.7), respectively.

In what follows, whenα= 1₂, we present an example showing that inequalities (2.8)– (2.11) are sharper than (1.4)–(1.7), respectively.

Example1 TakeA= 2 3 0

0 5₇

andB= 10

3 0

0 23₇

. We ﬁnd 1₂ <A< 3₄< 3 <B< 4. A calculation

shows K_S₍₂₎(8)≈2.3847 <K(8)≈2.5313.

3 Conclusions

In this paper, we have presented some new weighted arithmetic–geometric operator mean inequalities. These inequalities are reﬁnements and generalizations of some correspond-ing results of [1,2].

Acknowledgements

The author would like to express her sincere thanks to referees and editor for their enthusiastic guidance and help.

Funding

This research was supported by the Scientiﬁc Research Fund of Yunnan Provincial Education Department (Grant Nos. 2014Y645, 2018JS747).

Competing interests

The author declares that she has no competing interests.

Authors’ contributions

The author read and approved the ﬁnal manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 30 January 2018 Accepted: 25 June 2018 References

1. Lin, M.: Squaring a reverse AM-GM inequality. Stud. Math.215, 187–194 (2013)

2. Zhang, P.: More operator inequalities for positive linear maps. Banach J. Math. Anal.9, 166–172 (2015) 3. Tominaga, M.: Specht’s ratio in the Young inequality. Sci. Math. Jpn.55, 583–588 (2002)

4. Xue, J.: Some reﬁnements of operator reverse AM-GM mean inequalities. J. Inequal. Appl.2017, Article ID 283 (2017) 5. Zou, L.: An arithmetic-geometric mean inequality for singular values and its applications. Linear Algebra Appl.528,

25–32 (2017)

6. Hu, X.: Some inequalities for unitarily invariant norms. J. Math. Inequal.6, 615–623 (2012)

7. Bhatia, R., Kittaneh, F.: Notes on matrix arithmetic-geometric mean inequalities. Linear Algebra Appl.308, 203–211 (2000)

8. Bhatia, R.: Positive Deﬁnite Matrices. Princeton University Press, Princeton (2007)