Unitarily invariant norm inequalities for matrix means

ORIGINAL RESEARCH PAPER

Unitarily invariant norm inequalities for matrix means

Hongliang Zuo^{1,2 •}Fazhen Jiang³

Received: 22 July 2019 / Accepted: 11 November 2020 / Published online: 12 January 2021 ÓThe Author(s) 2021

Abstract

The main target of this article is to present several unitarily invariant norm inequalities which are refinements of arithmetic-geometric mean, Heinz and Cau- chy-Schwartz inequalities by convexity of some special functions.

Keywords Unitarily invariant norm inequalityYoung inequalityHeinz inequalityCauchy-Schwartz inequality

Mathematics Subject Classification Primary 39B82 Secondary 44B20 46C05

1 Introduction

In this sequel, we use the standard notationMn;M_n^þandM^þþ_n for the algebra of all nn complex matrices, the cone of positive (or positive semidefinite) matrix and that of strictly positive matrices inMn, respectively. Matrices and their inequalities have attracted researchers working in functional analysis. These inequalities have

Communicated by Samy Ponnusamy.

& Fazhen Jiang

FazhenJiang@163.com Hongliang Zuo zuodke@yahoo.com

1 School of Mathematics and Information Sciences, Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, Henan Normal University, Xinxiang 453007, Henan, People’s Republic of China

2 Key Laboratory of Applied Mathematics (Putian University), Fujian Province University, Putian 351100, Fujian, People’s Republic of China

3 College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, Henan, People’s Republic of China

https://doi.org/10.1007/s41478-020-00286-2

been studied in different approaches among which unitarily invariant norms inequalities are most popular. Recall that a unitarily invariant norm is a normk k defined onM_n satisfying the property kUAVk ¼ kAk for allA2M_n and unitaries U;V2Mn. The absolute value of a matrixA¼ ðaijÞis defined byjAj ¼ ðAAÞ¹⁼². The motivation behind this work starts with some crucial inequalities which will be presented as follows.

The classical arithmetic-geometric mean inequality [1] states that forA;B2M_n^þ andX2Mn,

kA¹²XB¹²k 1

2kAXþXBk: ð1Þ

Heinz inequality [1] is a refinement of inequality (1) which states that

kA¹²XB¹²k kA^tXB^1tþA^1tXB^t

2 k kAXþXB

2 k ð2Þ

hold forA;B2M_n^þ;X2M_n and 0t1.

A general form of Cauchy-Schwartz inequality [2] states that forA;B2M^þ_n;X2 M_n andr[0,

kjA¹²XB¹²j^rk² kjAXj^rkkjXBj^rk: ð3Þ We remark that the above inequalities have been studied deeply in the literature. We refer the reader to [3–5] as samples of recent work treating such inequalities and their variants.

Motivated by Bhatia and Bourin [2,6], here we define two functionsfandhfor a given unitarily invariant normk k,

fðtÞ ¼ kA^tXB^1tkkA^1tXB^tk and hðtÞ ¼ kA^tXB^1tþA^1tXB^t

2 k²;

whereA;B2M^þ_n andX2Mn. The above functionsfandhare convex on [0,1] and attain their minimum att¼¹₂. In this article, we utilize convexity of these functions to obtain refinements of arithmetic-geometric mean, Heinz and Cauchy-Schwartz inequalities. The following convex function inequalities are also essential to our results.

Hermite-Hadaward inequality [7] states that for every real-valued convex functiongon the interval [a,b], we have

g aþb 2

1

ba Z b

a

gðtÞdtgðaÞ þgðbÞ

2 :

In 2010, EL Farissi [8] refined Hermite-Hadaward inequality as follows

g aþb 2

lðkÞ 1 ba

Z b a

gðtÞdtLðkÞ gðaÞ þgðbÞ 2 for allk2 ½0;1, where

lðkÞ ¼kg kbþ ð2kÞa 2

þ ð1kÞg ð1þkÞbþ ð1kÞa 2

and

LðkÞ ¼1

2ðgðkbþ ð1kÞaÞ þkgðaÞ þ ð1kÞgðbÞÞ:

A few years later, Abbas and Mourad [9] got that

g aþb 2

1

ba Z b

a

gðtÞdt 1

4n ð2n1ÞgðaÞ þ2g aþb 2

þ ð2n1ÞgðbÞ

gðaÞ þgðbÞ

2 :

The following lemma combining Farissi and Abbas’ results will be essential for our main results. The main results in this paper, Theorems1,2and3, are obtained by applying some refinements of Hermite-Hadaward inequalities on the convex func- tionsfandhusing the same method from Kittaneh [10].

Lemma 1 Let g be a real-valued convex function which is convex on the interval [a,b]. Then for any positive integer n,we have

g aþb 2

1

ng bþ ð2n1Þa 2n

þ 11 n

g ðnþ1Þbþ ðn1Þa 2n

1

ba Z b

a

gðtÞdt 1

4n ð2n1ÞgðaÞ þ2g aþb 2

þ ð2n1ÞgðbÞ

gðaÞ þgðbÞ

2 :

Recently, Chen, Chen and Gao [11] obtained the following refinements of Hermite-Hadaward inequality.

Lemma 2 Let m;n:½a;b ! ½0;þ1Þ be convex functions and meet

½mðaÞ mðbÞ ½nðaÞ nðbÞ 0. Then for allk2 ½0;1,we have 1

ba Z b

a

mðtÞnðtÞdtL⁰ðkÞ 1

3Mða;bÞ þ1 6Nða;bÞ and

2m aþb 2

n aþb 2

1

6Mða;bÞ 1

3Nða;bÞ l⁰ðkÞ 1 ba

Z b a

mðtÞnðtÞdt where

Mða;bÞ ¼mðaÞnðaÞ þmðbÞnðbÞ;Nða;bÞ ¼mðaÞnðbÞ þmðbÞnðaÞ;

L⁰ðkÞ ¼k

3mðaÞnðaÞ þ1k

3 mðbÞnðbÞ þk

3mðkbþ ð1kÞaÞnðkbþ ð1kÞaÞ þk

6mðkbþ ð1kÞaÞ½knðaÞ þ ð1kÞnðbÞ þk

6nðkbþ ð1kÞaÞ½knðbÞ þ ð1kÞnðaÞ and

l⁰ðkÞ ¼2km ð2kÞaþkb 2

n ð2kÞaþkb 2

1þ3k3k² 6 Mða;bÞ 23kþ3k²

6 Nða;bÞ þ2ð1kÞm ð1kÞaþ ð1þkÞb 2

n ð1kÞaþ ð1þkÞb 2

:

The organization of this article will be as follows. In the following, we mainly present some unitarily invariant norm inequalities for matrix means which are refinements of arithmetic-geometric mean, Heinz and Cauchy-Schwartz inequalities utilizing Lemmas1and2.

2 Unitarily invariant norm inequalities Now we are in a position to begin our main results.

Applying Lemma1 to the convex functionh(t) on the interval½l;1lwhen 0l\¹₂ and on the interval ½1l;l when ¹₂\l1, we obtain the following refinement of arithmetic-geometric mean and Heinz inequalities.

Theorem 1 If A;B2M_n^þ,X2Mn and0l1,then for unitarily invariant norm k k,

kA¹²XB¹²k²1

nkA^{1þð2n2Þl2n} XB^{11þð2n2Þl2n} þA^{11þð2n2Þl2n} XB^{1þð2n2Þl2n}

2 k²

1

j2l1jj Z 1l

l

kA^tXB^1tþA^1tXB^t

2 k²dtj l6¼1 2

1

2n ð2n1ÞkA^lXB^1lþA^1lXB^l

2 k²þ kA¹²XB¹²k²

kA^lXB^1lþA^1lXB^l

2 k²

hold for any positive integern.

Proof Assume thatA;B2M^þ_n,X2M_nand 0l\¹₂, then it follows by Lemma1 that

h lþ1l 2

1

nh 1lþ ð2n1Þl 2n

þ 11 n

h ðnþ1Þð1lÞ þ ðn1Þl 2n

1 12l

Z 1l l

hðtÞdt l6¼1 2

1

4n ð2n1ÞhðlÞ þ2h lþ1l 2

þ ð2n1Þhð1lÞ

hðlÞ þhð1lÞ

2 ;

which is equivalent to

h 1

2 1

nh 1lþ ð2n1Þl 2n

þ 11 n

h ðnþ1Þð1lÞ þ ðn1Þl 2n

1

12l Z 1l

l

hðtÞdt l6¼1 2

1

4n ð2n1ÞhðlÞ þ2h 1

2 þ ð2n1Þhð1lÞ

hðlÞ þhð1lÞ

2 :

Hence,

kA¹²XB¹²k²1

nkA^{1þð2n2Þl2n} XB^{11þð2n2Þl2n} þA^{11þð2n2Þl2n} XB^{1þð2n2Þl2n}

2 k²

1

12l Z 1l

l

kA^tXB^1tþA^1tXB^t

2 k²dt l6¼1 2

1

2n ð2n1ÞkA^lXB^1lþA^1lXB^l

2 k²þ kA¹²XB¹²k²

kA^lXB^1lþA^1lXB^l

2 k²:

ð4Þ

On the other hand, if¹₂\l1, then it follows by symmetry (i.e., by applying the above inequality (4) to 1l) that

kA¹²XB¹²k²1

nkA^{1þð2n2Þl2n} XB^{11þð2n2Þl2n} þA^{11þð2n2Þl2n} XB^{1þð2n2Þl2n}

2 k²

1

2l1 Z l

1l

kA^tXB^1tþA^1tXB^t

2 k²dt l6¼1 2

1

2n ð2n1ÞkA^lXB^1lþA^1lXB^l

2 k²þ kA¹²XB¹²k²

kA^lXB^1lþA^1lXB^l

2 k²:

ð5Þ

We complete the proof of Theorem1by combining the inequalities (4) and (5).h Following the same logic of Theorem1and applying Lemma 1to the function f(t) on the interval ½l;1l when 0l\¹₂ and on the interval ½1l;l when

1

2\l1, we have the following refinement of Cauchy-Schwartz inequality.

Theorem 2 If A;B2M_n^þ,X2Mn and0l1,then for unitarily invariant norm k k,

kA¹²XB¹²k²1

nkA^{1þð2n2Þl2n} XB^{11þð2n2Þl2n} kkA^{11þð2n2Þl2n} XB^{1þð2n2Þl2n} k

1

j12ljj Z 1l

l

kA^tXB^1tkkA^1tXB^tkdtj l6¼1 2

1

2nhð2n1ÞkA^lXB^1lkkA^1lXB^lk þ kA¹²XB¹²k²i kA^lXB^1lkkA^1lXB^lk

hold for any positive integern.

Proof Assume thatA;B2M^þ_n,X2M_nand 0l\¹₂, then it follows by Lemma1 that

f lþ1l 2

1

nf 1lþ ð2n1Þl 2n

þ 11 n

f ðnþ1Þð1lÞ þ ðn1Þl 2n

1 12l

Z 1l l

fðtÞdt l6¼1 2

1

4n ð2n1ÞfðlÞ þ2f lþ1l 2

þ ð2n1Þfð1lÞ

fðlÞ þfð1lÞ

2 ;

which is equivalent to

f 1 2 1

nf 1lþ ð2n1Þl 2n

þ 11 n

f ðnþ1Þð1lÞ þ ðn1Þl 2n

1

12l Z 1l

l

fðtÞdt l6¼1 2

1

4n ð2n1ÞfðlÞ þ2f 1

2 þ ð2n1Þfð1lÞ

fðlÞ þfð1lÞ

2 :

Hence,

kA¹²XB¹²k²1

nkA^{1þð2n2Þl2n} XB^{11þð2n2Þl2n} kkA^{11þð2n2Þl2n} XB^{1þð2n2Þl2n} k

1

12l Z 1l

l

kA^tXB^1tkkA^1tXB^tkdt l6¼1 2

1

2nhð2n1ÞkA^lXB^1lkkA^1lXB^lk þ kA¹²XB¹²k²i kA^lXB^1lkkA^1lXB^lk:

ð6Þ

On the other hand, if¹₂\l1, then it follows by symmetry (i.e., by applying the above inequality (6) to 1l) that

kA¹²XB¹²k²1

nkA^{1þð2n2Þl2n} XB^{11þð2n2Þl2n} kkA^{11þð2n2Þl2n} XB^{1þð2n2Þl2n} k

1

2l1 Z l

1l

kA^tXB^1tkkA^1tXB^tkdt l6¼1 2

1

2nhð2n1ÞkA^lXB^1lkkA^1lXB^lk þ kA¹²XB¹²k²i kA^lXB^1lkkA^1lXB^lk:

ð7Þ

We complete the proof of Theorem2by combining the inequalities (6) and (7).h Next, for every positive real numberr, we consider the function

/ðtÞ ¼ kjA^tXB^1tj^rk kjA^1tXB^tj^rk

which is convex on [0,1] and attains its minimum att¼¹₂obtained by Hiai and Zhan [12].

Applying Lemma1to the function/ðtÞon the interval½l;1lwhen 0l\¹₂ and on the interval½1l;lwhen¹₂\l1, then we have the following refinement of general Cauchy-Schwartz inequality.

Theorem 3 If A;B2M_n^þ,X2Mn and0l1,then for unitarily invariant norm k k,

kjA¹²XB¹²j^rk²1

nkjA^{1þð2n2Þl2n} XB^{11þð2n2Þl2n} j^rk kjA^{11þð2n2Þl2n} XB^{1þð2n2Þl2n} j^rk

1

j12ljj Z 1l

l

kjA^tXB^1tj^rk kjA^1tXB^tj^rkdtj l6¼1 2

1

2nhð2n1ÞkjA^lXB^1lj^rk kjA^1lXB^lj^rk þ kjA¹²XB¹²j^rk²i kjA^lXB^1lj^rk kjA^1lXB^lj^rk

hold for any positive integern.

Proof Assume thatA;B2M^þ_n,X2M_nand 0l\¹₂, then it follows by Lemma1 that

/ lþ1l 2

1

n/ 1lþ ð2n1Þl 2n

þ 11 n

/ ðnþ1Þð1lÞ þ ðn1Þl 2n

1 12l

Z 1l l

/ðtÞdt l6¼1 2

1

4n ð2n1Þ/ðlÞ þ2/ lþ1l 2

þ ð2n1Þ/ð1lÞ

/ðlÞ þ/ð1lÞ

2 ;

which is equivalent to

/ 1 2 1

n/ 1lþ ð2n1Þl 2n

þ 11 n

/ ðnþ1Þð1lÞ þ ðn1Þl 2n

1

12l Z 1l

l

/ðtÞdt l6¼1 2

1

4n ð2n1Þ/ðlÞ þ2/ð1

2Þ þ ð2n1Þ/ð1lÞ

/ðlÞ þ/ð1lÞ

2 :

Hence,

kjA¹²XB¹²j^rk²1

nkjA^{1þð2n2Þl2n} XB^{11þð2n2Þl2n} j^rk kjA^{11þð2n2Þl2n} XB^{1þð2n2Þl2n} j^rk

1

12l Z 1l

l

kjA^tXB^1tj^rk kjA^1tXB^tj^rkdt l6¼1 2

1

2nhð2n1ÞkjA^lXB^1lj^rk kjA^1lXB^lj^rk þ kjA¹²XB¹²j^rk²i kjA^lXB^1lj^rk kjA^1lXB^lj^rk:

ð8Þ

On the other hand, if¹₂\l1, then it follows by symmetry (i.e., by applying the above inequality (8) to 1l) that

kjA¹²XB¹²j^rk²1

nkjA^{1þð2n2Þl2n} XB^{11þð2n2Þl2n} j^rk kjA^{11þð2n2Þl2n} XB^{1þð2n2Þl2n} j^rk

1

2l1 Z l

1l

kjA^tXB^1tj^rk kjA^1tXB^tj^rkdt l6¼1 2

1

2nhð2n1ÞkjA^lXB^1lj^rk kjA^1lXB^lj^rk þ kjA¹²XB¹²j^rk²i kjA^lXB^1lj^rk kjA^1lXB^lj^rk:

ð9Þ

We complete the proof of Theorem3by combining the inequalities (8) and (9).h In view of the fact that the functionsf(t) andh(t) are symmetric, we have

jfðlÞ fð1lÞj jhðlÞ hð1lÞj

¼ jkA^lXB^1lkkA^1lXB^lk kA^1lXB^lkkA^lXB^1lkj

¼0:

We can have the following result by applying Lemma2 to function

fðtÞ hðtÞ ¼ kA^tXB^1tkkA^1tXB^tkkA^lXB^1lþA^1lXB^l

2 k²:

Corollary 1 For 0l1and allk2 ½0;1,we have

2kA¹²XB¹²k⁴ kA^lXB^1lkkA^1lXB^lkkA^lXB^1lþA^1lXB^l

2 k²

l⁰ðkÞ

1

j12ljj Z 1l

l

kA^tXB^1tkkA^1tXB^tkkA^tXB^1tþA^1tXB^t

2 k²dtj l6¼1 2

L⁰ðkÞ

kA^lXB^1lkkA^1lXB^lkkA^lXB^1lþA^1lXB^l

2 k²;

where

L⁰ðkÞ ¼1

3kA^1lXB^lkkA^lXB^1lkkA^1lXB^lþA^lXB^1l

2 k²

þk

3kA^kþl2klXB^1ðkþl2klÞkkA^1ðkþl2klÞXB^kþl2klk kA^kþl2klXB^1ðkþl2klÞþA^1ðkþl2klÞXB^kþl2kl

2 k²

þk

6kA^kþl2klXB^1ðkþl2klÞkkA^1ðkþl2klÞXB^kþl2klk kA^lXB^1lþA^1lXB^l

2 k²

þk

6kA^kþl2klXB^1ðkþl2klÞþA^1ðkþl2klÞXB^kþl2kl

2 k²

kA^1lXB^lþA^lXB^1l

2 k²

and

l⁰ðkÞ ¼2kkA^kþ2l2kl2 XB^1kþ2l2kl2 kkA^1kþ2l2kl2 XB^kþ2l2kl2 k kA^kþ2l2kl2 XB^1kþ2l2kl2 þA^1kþ2l2kl2 XB^kþ2l2kl2

2 k²

kA^lXB^1lkkA^1lXB^lkkA^lXB^1lþA^1lXB^l

2 k²

þ2ð1kÞkA^kþ12kl2 XB^1kþ12kl2 kkA^1kþ12kl2 XB^kþ12kl2 k kA^kþ12kl2 XB^1kþ12kl2 þA^1kþ12kl2 XB^kþ12kl2

2 k²:

Here we remark that jfðlÞ fð1lÞj j/ðlÞ /ð1lÞj ¼0 and jhðlÞ hð1lÞj j/ðlÞ /ð1lÞj ¼0 for 0l1. Hence, results similar to Corollary 1can be obtained by usingf /andh/.

AcknowledgementsWe thank the referee for careful review and valuable comments. This research was supported in part by Key Laboratory of Applied Mathematics of Fujian Province University (Putian University, No. SX201901) and in part by the 2018 Scientific Research Project for Postgraduates of Henan Normal University (Approved Document Number: [2018] 2). This work is funded by National Natural Science Foundation of China with Grant number 11501176.

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Conflict of interest The authors declare that they have no conflict of interest.

Ethical approval This article does not contain any studies with animals performed by any of the authors.

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Affiliations

Hongliang Zuo^{1,2 •}Fazhen Jiang³

1 School of Mathematics and Information Sciences, Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, Henan Normal University, Xinxiang 453007, Henan, People’s Republic of China

2 Key Laboratory of Applied Mathematics (Putian University), Fujian Province University, Putian 351100, Fujian, People’s Republic of China

3 College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, Henan, People’s Republic of China