On further refinements for Young inequalities

Shigeru Furuichi; Hamid Reza Moradi

doi:10.1515/math-2018-0115

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On further refinements for Young inequalities

Shigeru Furuichi and Hamid Reza Moradi

Published/Copyright: December 27, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

In this paper sharp results on operator Young’s inequality are obtained. We first obtain sharp multiplicative refinements and reverses for the operator Young’s inequality. Secondly, we give an additive result, which improves a well-known inequality due to Tominaga. We also provide some estimates for the difference A^1/2(A^−1/2BA^−1/2)^vA^1/2-{(1-v)A + vB} for v∉[ 0,1].

Keywords: Operator inequality; Young inequality; Weighted arithmetic and geometric mean; Positive operator

MSC 2010: 47A63; 26D07; 47A60

1 Introduction

This note lies in the scope of operator inequalities. We assume that the reader is familiar with the continuous functional calculus and the Kubo-Ando theory [1].

It is to be understood throughout the paper that the capital letters present bounded linear operators acting on a Hilbert space 𝓗. A is positive (written A ≥ 0) in case 〈Ax,x〉 ≥ 0 for all x ∈ 𝓗 also an operator A is said to be strictly positive (denoted by A > 0) if A is positive and invertible. If A and B are self-adjoint, we write B ≥ A in case B–A ≥ 0. As usual, by I we denote the identity operator.

The weighted arithmetic mean ∇_v, geometric mean ♯_v, and harmonic mean !_v, for v ∈ [0, 1] and a,b > 0, are defined as follows:

a∇vb=1−va+vb,a♯vb=a1−vbv,a!vb=(1−v)a−1+vb−1−1.

If v = 12, we denote the arithmetic, geometric, and harmonic means, respectively, by ∇, ♯ and !, for the simplicity. Like the scalar cases, the operator arithmetic mean, the operator geometric mean, and the operator harmonic mean for A,B > 0 can be stated in the following form:

A∇vB=1−vA+vB,A♯vB=A12A−12BA−12vA12,A!vB=(1−v)A−1+vB−1−1.

The celebrated arithmetic-geometric-harmonic-mean inequalities for scalars assert that if a,b > 0, then

a!vb≤a♯vb≤a∇vb.(1)

Generalization of the inequalities (1) to operators can be seen as follows: If A,B > 0, then

A!vB≤A♯vB≤A∇vB.

The last inequality above is called the operator Young inequality. During the past years, several refinements and reverses were given for Young’s inequality, see for example [2,3,4].

Zuo et al. showed in [5, Theorem 7] that the following inequality holds:

Kh,2rA♯vB≤A∇vB, r=minv,1−v, Kh,2=h+124h, h=Mm(2)

whenever 0 < m′I ≤ B ≤ mI < MI ≤ A ≤ M′I or 0 < m′I ≤ A ≤ mI < MI ≤ B ≤ M′I. As the authors mentioned in [5], the inequality (2) improves the following refinement of Young’s inequality involving Specht St=t1t−1elogt1t−1t>0,t≠1 (see [6, Theorem 2]),

ShrA♯vB≤A∇vB.

Under the above assumptions, Dragomir proved in [7, Corollary 1] that

A∇vB≤expv1−v2h−12A♯vB.

We remark that there is no relationship between two constants K(h,2)^r and expv1−v2h−12 in general.

In [3, 8] we proved some sharp multiplicative reverses of Young’s inequality. In this brief note, as the continuation of our previous works, we establish sharp bounds for the arithmetic, geometric and harmonic mean inequalities. Moreover, we shall show some additive-type refinements and reverses of Young’s inequality. We will formulate our new results in a more general setting, namely the sandwich assumption sA ≤ B ≤ tA (0 < s ≤ t). Additionally, we present some Young type inequalities for the wider range of v; i.e., v ∉ [0, 1].

2 Main results

In our previous work [8], we gave new sharp inequalities for reverse Young inequalities. In this section we firstly give new sharp inequalities for Young inequalities, as limited cases in the first inequalities both (i) and (ii) of the following theorem.

Theorem 2.1

Let A,B > 0 such that sA ≤ B ≤ tA for some scalars 0 < s ≤ t and let f_v(x) ≡(1−v)+vxxvforx>0,and v ∈ [0, 1].

If t ≤ 1, then f_v(t) A ♯_vB ≤ A∇_vB ≤ f_v(s) A♯_vB.
If s ≥ 1, then f_v(s) A♯_vB ≤ A∇_vB ≤ f_v(t)A♯_vB.

Proof

Since fv′(x) = v(1–v)(x–1)x^−v-1, f_v(x) is monotone decreasing for 0 < x ≤ 1 and monotone increasing for x ≥ 1.

For the case 0 < s ≤ x ≤ t ≤ 1, we have f_v(t) ≤ f_v(x) ≤ f_v(s), which implies f_v(t) A ♯_vB ≤ A∇_vB ≤ f_v(s) A♯_vB by the standard functional calculus.□
For the case 1 ≤ s ≤ x ≤ t, we have f_v(s) ≤ f_v(x) ≤ f_v(t) which implies f_v(s) A♯_vB ≤ A∇_vB ≤ f_v(t)A♯_vB by the standard functional calculus.

Remark 2.2

It is worth emphasizing that each assertion in Theorem 2.1 implies the other one. For instance, assume that the assertion (ii) holds, i.e.,

fvs≤fvx≤fvt, 1≤s≤x≤t.(3)

Let t ≤ 1, then1≤1t≤1x≤1s.Hence(3)ensures that

fv1t≤fv1x≤fv1s.

1−vt+vt1−v≤1−vx+vx1−v≤1−vs+vs1−v.

Now, by replacing v by 1–v we get

1−v+vttv≤1−v+vxxv≤1−v+vssv

which means

fvt≤fvx≤fvs, 0<s≤x≤t≤1.

In the same spirit, we can derive (ii) from (i).

Corollary 2.3

Let A, B > 0, m, m′, M, M′ > 0, and v ∈ [0, 1].

If 0 < m′I ≤ A ≤ mI < MI ≤ B ≤ M′I, then
m∇vMm♯vMA♯vB≤A∇vB≤m′∇vM′m′♯vM′A♯vB.(4)
If 0 < m′I ≤ B ≤ mI < MI ≤ A ≤ M′I, then
M∇vmM♯vmA♯vB≤A∇vB≤M′∇vm′M′♯vm′A♯vB.(5)

Proof

We use again the function fv(x)=(1−v)+vxxvin this proof.

The condition (i) is equivalent to I≤MmI≤A−12BA−12≤M′m′I, so that we get fv(Mm)A♯vB≤A∇vB≤fv(M′m′)A♯vB by putting s=Mmandt=M′m′ in (ii) of Theorem 2.1.

Similarly, the condition (ii) is equivalent to m′M′I≤A−12BA−12≤mMI≤I, so that we get fv(mM)A♯vB≤A∇vB≤fv(m′M′)A♯vB by putting s=m′M′andt=mM of Theorem 2.1.□

Note that the second inequalities in both (i) and (ii) of Theorem 2.1 and Corollary 2.3 are special cases of [8, Theorem A].

Remark 2.4

It is remarkable that the inequalities f_v(t) ≤ f_v(x) ≤ f_v(s) (0 < s ≤ x ≤ t ≤ 1) given in the proof of Theorem 2.1 are sharp, since the function f_v(x) for s ≤ x ≤ t is continuous. So, all results given from Theorem 2.1 are similarly sharp. As a matter of fact, let A = MI and B = mI, then from LHS of(5), we infer

A∇vB=(M∇vm)I and A♯vB=(M♯vm)I.

Consequently,

M∇vmM♯vmA♯vB=A∇vB.

To see that the constantm∇vMm♯vMin the LHS of(4)can not be improved, we consider A = mIand B = MI, then

m∇vMm♯vMA♯vB=A∇vB.

By replacing A, B by A^–1, B^–1, respectively, the refinement and reverse of non-commutative geometric-harmonic mean inequality can be obtained as follows:

Corollary 2.5

Let A, B > 0, m, m′, M, M′ > 0, and v ∈ [0, 1].

If 0 < m′I ≤ A ≤ mI < MI ≤ B ≤ M′I, then
m′!vM′m′♯vM′A♯vB≤A!vB≤m!vMm♯vMA♯vB.
If 0 < m′I ≤ B ≤ mI < MI ≤ A ≤ M′I, then
M′!vm′M′♯vm′A♯vB≤A!vB≤M!vmM♯vmA♯vB.

Now, we give a new sharp reverse inequality for Young’s inequality as an additive-type in the following.

Theorem 2.6

Let A,B > 0 such that sA ≤ B ≤ tA for some scalars 0 < s ≤ t, and v ∈ [0, 1]. Then

A∇vB−A♯vB≤maxgvs,gvtA(6)

where g_v(x )≡ (1–v) + vx–x^vfor s ≤ x ≤ t.

Proof

Straightforward differentiation shows that gv″ (x) = v(1–v)x^v–2 ≥ 0 and g_v(x) is continuous on the interval [s,t], so

gvx≤maxgvs,gvt.

Therefore, by applying similar arguments as in the proof of Theorem 2.1, we reach the desired inequality (6). This completes the proof of theorem.□

Corollary 2.7

Let A,B > 0 such that mI ≤ A,B ≤ MI for some scalars 0 < m < M. Then for v ∈ [0, 1],

A∇vB−A♯vB≤ξA

whereξ=max1MM∇vm−M♯vm,1mm∇vM−m♯vM.

Remark 2.8

We claim that if A,B > 0 such that mI ≤ A,B ≤ MI for some scalars 0 < m < M with h = Mmand v ∈ [0, 1], then

A∇vB−A♯vB≤maxgvh,gv1hA≤L1,hlog⁡ShA

holds, whereLx,y=y−xlog⁡y−log⁡xx≠yis the logarithmic mean and the term S(h) refers to the Specht’s ratio. Indeed, we have the inequalities

(1−v)+vh−hv≤L(1,h)log⁡S(h),(1−v)+v1h−h−v≤L(1,h)log⁡S(h),

which were originally proved in [9, Lemma 3.2], thanks toSh=S1handL1,h=L1,1h.Therefore, our result, Theorem 2.6, improves the well-known result by Tominaga [9, Theorem 3.1],

A∇vB−A♯vB≤L1,hlog⁡ShA.

Since g_v(x) is convex, we can not obtain a general result on the lower bound for A∇_vB–A♯_vB. However, if we impose the conditions, we can obtain new sharp inequalities for Young inequalities as an additive-type in the first inequalities both (i) and (ii) in the following proposition. (At the same time, of course, we also obtain the upper bounds straightforwardly.)

Proposition 2.9

Let A,B > 0 such that sA ≤ B ≤ tA for some scalars 0 < s ≤ t, v ∈ [0, 1], and g_v is defined as in Theorem 2.6.

If t ≤ 1, then g_v(t)A ≤ A∇_vB–A♯_vB ≤ g_v(s)A.
If s ≥ 1, then g_v(s)A ≤ A∇_vB–A♯_vB ≤ g_v(t)A.

Proof

It follows from the fact that g_v(x) is monotone decreasing for 0 < x ≤ 1 and monotone increasing for x ≥ 1.□

Corollary 2.10

Let A, B > 0, m, m′, M, M′ > 0, and v ∈ [0, 1].

If 0 < m′I ≤ A ≤ mI < MI ≤ B ≤ M′I, then
1mm∇vM−m♯vMA≤A∇vB−A♯vB≤1m′m′∇vM′−m′♯vM′A.
If 0 < m′I ≤ B ≤ mI < MI ≤ A ≤ M′I, then
1MM∇vm−M♯vmA≤A∇vB−A♯vB≤1M′M′∇vm′−M′♯vm′A.

In the following we use the notations ▽_v and ♮_v to distinguish from the operator means ∇_v and ♯_v:

A▽vB=1−vA+vB,A♮vB=A12A−12BA−12vA12

for v ∉ [0, 1]. Notice that, since A,B > 0, the expressions A▽_vB and A♮_vB are also well-defined.

Remark 2.11

It is known (and easy to show) that for any A,B > 0,

A▽vB≤A♮vB,forv∉0,1.

Assume g_v(x) is defined as in Theorem 2.6. By an elementary computation we have

gv′x>0forv∉0,1and0<x≤1gv′x<0forv∉0,1andx>1.

Now, in the same way as above we have also for any v ∉ [0, 1]:

If 0 < m′I ≤ A ≤ mI ≤ MI ≤ B ≤ M′I, then
1m′m′♮vM′−m′▽vM′A≤A♮vB−A▽vB≤1mm♮vM−m▽vMA.
On account of assumptions, we also infer
(m′♮vM′−m′▽vM′)I≤A♮vB−A▽vB≤(m♮vM−m▽vM)I.
If 0 < m′I ≤ B ≤ mI ≤ MI ≤ A ≤ M′I, then
1MM♮vm−M▽vmA≤A♮vB−A▽vB≤1M′M′♮vm′−M′▽vm′A.
On account of assumptions, we also infer
(M♮vm−M▽vm)I≤A♮vB−A▽vB≤(M′♮vm′−M′▽vm′)I.

In addition, with the same assumption to Theorem 2.6 except for v ∉ [0, 1], we have

min{gv(s),gv(t)}A≤A▽vB−A♮vB,

since we have min{g_v(s),g_v(t)} ≤ g_v(x) bygv″ (x) ≤ 0, for v ∉[0, 1].

Acknowledgement

The authors thank anonymous referees for giving valuable comments and suggestions to improve our manuscript. The author (S.F.) was partially supported by JSPS KAKENHI Grant Number 16K05257.

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Received: 2018-07-21

Accepted: 2018-10-18

Published Online: 2018-12-27

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https://doi.org/10.1515/math-2018-0115

Keywords for this article

Operator inequality; Young inequality; Weighted arithmetic and geometric mean; Positive operator

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