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Bounds of the logarithmic mean
Journal of Inequalities and Applications volume 2013, Article number: 535 (2013)
Abstract
We give tight bounds for the logarithmic mean. We also give new Frobenius norm inequalities for two positive semidefinite matrices. In addition, we give some matrix inequalities on the matrix power mean.
MSC: 15A39, 15A45.
1 Introduction
In this short paper, we study the bounds of the logarithmic mean which is defined by
for two positive numbers a and b. (We conventionally define $L(a,b)=a$ if $a=b$.) In the paper [1], the following relations were shown:
We now have the following lemma.
Lemma 1.1 For $a,b>0$, we have
Proof The second inequality of (3) can be proven easily. Indeed, we put $f(t)\equiv \frac{2}{3}{t}^{3}+\frac{1}{6}(1+{t}^{6})\frac{1}{8}{(1+{t}^{2})}^{3}$. Then we have ${f}^{\prime}(t)<0$ for $0<t<1$ and ${f}^{\prime}(t)>0$ for $t>1$. Thus we have $f(t)\ge f(1)=0$ for $t>0$. □
The first inequality of (3) refines the inequality
which is known as the classical Pólya inequality [2, 3].
Throughout this paper, we use the notation $M(n,\mathbb{C})$ as the set of all $n\times n$ matrices on the complex field ℂ. We also use the notation ${M}_{+}(n,\mathbb{C})$ as the set of all $n\times n$ positive semidefinite matrices. Here, $A\in {M}_{+}(n,\mathbb{C})$ means we have $\u3008\varphi A\varphi \u3009\ge 0$ for any vector $\varphi \u3009\in {\mathbb{C}}^{n}$. For $A\in M(n,\mathbb{C})$, the Frobenius norm (HilbertSchmidt norm) ${\parallel \cdot \parallel}_{F}$ is defined by
In the paper [2], the following norm inequality was shown.
Theorem 1.2 ([2])
For $A,B\in {M}_{+}(n,\mathbb{C})$, $X\in M(n,\mathbb{C})$ and the Frobenius norm ${\parallel \cdot \parallel}_{F}$, we have
From Lemma 1.1, we have the following proposition.
Proposition 1.3 For $A,B\in {M}_{+}(n,\mathbb{C})$, $X\in M(n,\mathbb{C})$ and the Frobenius norm ${\parallel \cdot \parallel}_{F}$, we have
To the first author’s best knowledge, the first inequality in Proposition 1.3 was suggested in [4].
This proposition can be proven in a similar way to the proof of Theorem 1.2 (or the proof of Theorem 2.2 which will be given in the next section), and this refines inequality (6) shown in [2].
2 Lower bound of the logarithmic mean
The following inequalities were given in [5]. Hiai and Kosaki gave the norm inequalities for Hilbert space operators in [5]. See also [6, 7]. Here we give them as a matrix setting to unify the description of this paper.
Theorem 2.1 ([5])
For $A,B\in {M}_{+}(n,\mathbb{C})$, $X\in M(n,\mathbb{C})$, $m\ge 1$ and every unitarily invariant norm $\u2980\cdot \u2980$, we have
The Frobenius norm is one of unitarily invariant norms. We give the refinement of the lower bound of the first inequality above for the Frobenius norm. That is, we have the following inequalities.
Theorem 2.2 For $A,B\in {M}_{+}(n,\mathbb{C})$, $X\in M(n,\mathbb{C})$, $m\ge 1$ and the Frobenius norm ${\parallel \cdot \parallel}_{F}$, we have
To prove Theorem 2.2, we need a few lemmas.
Lemma 2.3 Let u, v, w be nonnegative integers such that $w\ge u$, and let x be a positive real number. Then we have
Proof It is trivial for the case $x=1$ or $v=0$. We prove for the case $x\ne 1$ and $v\ne 0$. In addition, for the case that $u=w$, the equality holds. Thus we may assume $w>u$ and $v\ge 1$. Then the lemma can be proven in the following way.
□
Lemma 2.4 For a positive real number x and a natural number m, we have
Proof For the case $x=1$, the equality holds. So we prove this lemma for $x\ne 1$. If m is an odd number, then we have $m=2\lfloor m/2\rfloor +1$. Since we then have $m+1=2(\lfloor m/2\rfloor +1)$ and $2(\lfloor m/2\rfloor +1)1=m$, we have
If we put $\tilde{k}=\lfloor m/2\rfloor +1$, then the above means $(2\tilde{k}1)(m+1)=2\tilde{k}m$. Then the difference of the $\tilde{k}$th term of the both sides in inequality (9) is equal to 0. For the case that m is an even number, it never happens that the difference of the $\tilde{k}$th term of the both sides in inequality (9) is equal to 0. Therefore we have
where ${a}_{l}=(2l1)(m+1)$, ${b}_{l}=m(2l1)$ and ${c}_{l}=2\{m(l1)\}m$ for $l=1,2,\dots ,\lfloor m/2\rfloor $. Here we have ${c}_{l}{a}_{l}=\{m(2l1)\}(2m+1)\ge 0$, whenever ${b}_{l}\ge 0$. By Lemma 2.3, if ${b}_{l}\ge 0$, then we have ${x}^{{a}_{l}}(1{x}^{{b}_{l}})+{x}^{{c}_{l}}({x}^{{b}_{l}}1)\ge 0$. Thus the proof of this lemma is completed. □
If we put $t={x}^{2m(m+1)}>0$, then we have
which implies
We then have the following lemma.
Lemma 2.5 For $a,b>0$ and $m\ge 1$, we have
Proof The second inequality follows by inequality (11). We use the famous inequality $(x1)/logx\ge \sqrt{x}$ for $x>0$. We put $x={t}^{1/m}$ in this inequality. Then we have, for $t>0$,
which implies the first inequality. The third inequality can be proven by the use of the arithmetic meangeometric mean inequality. Thus the proof of this lemma is completed. □
We give some basic properties of the righthand side of inequality (13) in the Appendix.
Proof of Theorem 2.2 It is known that the Frobenius norm inequality immediately follows from the corresponding scalar inequality [6]. However, we give here an elementary proof for the convenience of the readers. Let ${U}^{\ast}AU={D}_{1}=diag\{{\alpha}_{1},\dots ,{\alpha}_{n}\}$ and ${V}^{\ast}BV={D}_{2}=diag\{{\beta}_{1},\dots ,{\beta}_{n}\}$. Then ${\alpha}_{1},\dots ,{\alpha}_{n}\ge 0$ and ${\beta}_{1},\dots ,{\beta}_{n}\ge 0$. We put ${U}^{\ast}XV\equiv Y=({y}_{ij})$. By the first inequality of (12), we have
Applying inequality (11), we have the second inequality of (7) in a similar way. The third inequality holds due to Theorem 2.1 (or the third inequality of (12)). □
3 Upper bound of the logarithmic mean
In the paper [5], the following norm inequalities were also given for Hilbert space operators. Here we give them for matrices, as we mentioned in the beginning of Section 2.
Theorem 3.1 ([5])
For $A,B\in {M}_{+}(n,\mathbb{C})$, $X\in M(n,\mathbb{C})$, $m\ge 2$ and every unitarily invariant norm $\u2980\cdot \u2980$, we have
We also give an improved upper bound of the logarithmic mean on Theorem 3.1 above, only for the Frobenius norm. Namely, we can prove the following inequalities in a similar way to the proof of Theorem 2.2, by the use of scalar inequalities which will be given in Lemma 3.4.
Theorem 3.2 For $A,B\in {M}_{+}(n,\mathbb{C})$, $X\in M(n,\mathbb{C})$, $m\ge 2$ and the Frobenius norm ${\parallel \cdot \parallel}_{F}$, we have
To prove Theorem 3.2, we need to prove the following lemmas.
Lemma 3.3 For $x>0$ and $m\ge 2$, we have
Proof For $m\ge 2$, we calculate
Here, we put $y\equiv {x}^{1/2}>0$, then we have
where ${p}_{l}=(2l1)m$, ${q}_{l}=2(m1)(ml)$ and ${r}_{l}=m2l$. The last inequality follows from Lemma 2.3, because we have ${q}_{l}{p}_{l}=(2m1)(m2l)\ge $ whenever ${r}_{l}\ge 0$. □
Lemma 3.4 For $a,b>0$ and $m\ge 2$, we have
Proof To prove the first inequality, we have only to prove the following inequality:
Inequality (19) can be proven by putting $x\equiv {t}^{1/m}>0$ in the famous inequality $(x1)/logx\le (x+1)/2$ for $x>0$.
To prove the second inequality of (18), it is sufficient to prove inequality (15) which follows from Lemma 3.3. We obtain actually the second inequality of (18) by putting $t={x}^{m(m1)}>0$ in inequality (15), and then putting $t=a/b$.
To prove the third inequality of (18), it is sufficient to prove the following inequality:
This inequality can be proven by the induction on m. Indeed, we assume that the inequality (20) holds for some m. Then we add ${t}^{m}>0$ to both sides of inequality (20). Then we have
Therefore we have only to prove the inequality
which is equivalent to the inequality
We put ${f}_{m}(t)\equiv (m1){t}^{m}m{t}^{m1}+1$. Then we can prove ${f}_{m}(t)\ge {f}_{m}(1)=0$ by elementary calculations. Thus inequality (20) holds for $m+1$. □
We give some basic properties of the righthand side of inequality (19) in the Appendix.
4 Matrix inequalities on the geometric mean
Using Lemma 1.1, Lemma 2.5 and Lemma 3.4, we have following Proposition 4.1, Proposition 4.2 and Proposition 4.3, respectively.
Proposition 4.1 For $A,B\in {M}_{+}(n,\mathbb{C})$, we have
where $A{\mathrm{\#}}_{\nu}B\equiv {A}^{1/2}{({A}^{1/2}B{A}^{1/2})}^{\nu}{A}^{1/2}$ ($0\le \nu \le 1$) is νweighted geometric mean introduced in [8].
Proposition 4.2 For $A,B\in {M}_{+}(n,\mathbb{C})$ and $m\ge 1$, we have
Proposition 4.3 For $A,B\in {M}_{+}(n,\mathbb{C})$ and $m\ge 2$, we have
We give the proof of Proposition 4.3. Proposition 4.1 and Proposition 4.2 are also proven in a similar way, using Lemma 1.1 and Lemma 2.5. In addition, by using the notion of the representing function ${f}_{m}(x)=1mx$ for the operator mean m, it is well known [8] that ${f}_{m}(x)\le {f}_{n}(x)$ holds for $x>0$ if and only if $AmB\le AnB$ holds for all positive operators A and B. However, we give an elementary proof for the convenience of the readers.
Proof of Proposition 4.3 Since $T\equiv {A}^{1/2}B{A}^{1/2}\ge 0$, there exists a unitary matrix U such that ${U}^{\ast}TU=D\equiv diag\{{\lambda}_{1},\dots ,{\lambda}_{n}\}$. Then ${\lambda}_{1},\dots ,{\lambda}_{n}\ge 0$. From Lemma 3.4, for $i=1,\dots ,n$, we have
Thus we have
Multiplying U and ${U}^{\ast}$ to both sides, we have
Inserting $T\equiv {A}^{1/2}B{A}^{1/2}$ and then multiplying two ${A}^{1/2}$ to all sides from both sides, we obtain the result. □
Closing this section, we give another matrix inequalities by the use of another lower bound of the logarithmic mean. As another lower bound of the logarithmic mean, the following inequalities are known:
The proofs of the above inequalities are not so difficult (they can be done by putting $x={t}^{1/3}>0$ and $x={t}^{1/6}>0$), here we omit them. From inequalities (21), we have
Inequalities (22) imply the following result in a similar way to the proof of Proposition 4.3.
Proposition 4.4 For $A,B\in {M}_{+}(n,\mathbb{C})$, we have
5 Comments
Proposition 6.2 given in the Appendix shows that our upper bound is tighter than the standard upper bound for the case $t>1$ and our lower bound is tighter than the standard lower bound for any $t>0$. In addition, our lower bound ${\alpha}_{m}(t)$ of the logarithmic mean $L(t,1)$ is tighter than the lower bound $\sqrt{t}$ given by Lin in [1] for $m\ge 1$. However, it may be a difficult problem to find the minimum $m\in \mathbb{N}$ such that ${\beta}_{m}(t)\le {(({t}^{1/3}+1)/2)}^{3}$ for any $t>0$. The righthand side of the above inequality is the upper bound given by Lin in [1]. (See inequalities (2).)
Appendix
Here we note some basic properties of the following scalar sums:
for $t>0$.
Proposition 6.1 For any $t>0$, we have following properties:

(i)
${\alpha}_{m}(t)\le {\alpha}_{m+1}(t)$.

(ii)
${\beta}_{m+1}(t)\le {\beta}_{m}(t)$.

(iii)
${\alpha}_{m}(t)$ and ${\beta}_{m}(t)$ converges to $L(t,1)$ as $m\to \mathrm{\infty}$. In addition, we have ${\alpha}_{m}(t)\le {\beta}_{m}(t)$.
Proof We prove (i)(iii) for $t\ne 1$, since it is trivial for the case $t=1$.

(i)
Since
$${\alpha}_{m}(t)=\frac{{t}^{1/(2m)}(t1)}{m({t}^{1/m}1)},$$
for $t\ne 1$, we prove
for $t>0$ and $t\ne 1$. We first prove the case $t>1$. Then we put $s\equiv {t}^{1/(2m(m+1))}$ and
By elementary calculations, we have ${f}_{m}(s)>{f}_{m}(1)=0$, which implies
We can prove similarly
for the case $0<s<1$.

(ii)
Since
$${\beta}_{m}(t)=\frac{({t}^{1/m}+1)(t1)}{2m({t}^{1/m}1)},$$
for $t\ne 1$, we prove
for $t>0$ and $t\ne 1$. We first prove the case $t>1$. Then we put $r\equiv {t}^{1/(m(m+1))}$ and
Since
we have ${g}_{m}(r)>{g}_{m}(1)=0$, which implies
We can prove similarly
for the case $0<r<1$.

(iii)
Since we have
$$\underset{m\to \mathrm{\infty}}{lim}m({t}^{1/m}1)=\underset{p\to 0}{lim}\frac{{t}^{p}1}{p}=logt$$
for $t>0$ and $t\ne 1$, we have
The arithmeticgeometric mean inequality proves ${\alpha}_{m}(t)<{\beta}_{m}(t)$ for $t>0$ and $t\ne 1$. □
As standard bounds of the Riemann sum for the integral ${\int}_{0}^{1}{t}^{\nu}\phantom{\rule{0.2em}{0ex}}d\nu $, we have
and
where
Then we have the following relations.
Proposition 6.2

(i)
For $0<t<1$, we have ${\alpha}_{m}(t)>{\gamma}_{m}(t)$ and ${\beta}_{m}(t)>{\delta}_{m}(t)$.

(ii)
For $t>1$, we have ${\alpha}_{m}(t)>{\delta}_{m}(t)$ and ${\beta}_{m}(t)<{\gamma}_{m}(t)$.
Proof

(i)
For the case $0<t<1$, the following calculations show assertion:
$${\alpha}_{m}(t){\gamma}_{m}(t)=\frac{(t1)({t}^{1/(2m)}{t}^{1/m})}{m({t}^{1/m}1)}>0,\phantom{\rule{2em}{0ex}}{\beta}_{m}(t){\delta}_{m}(t)=\frac{t+1}{2m}>0.$$ 
(ii)
For the case $t>1$, the following calculations show assertion:
$${\alpha}_{m}(t){\delta}_{m}(t)=\frac{t+{t}^{1/(2m)}}{m({t}^{1/(2m)}+1)}>0,\phantom{\rule{2em}{0ex}}{\beta}_{m}(t){\gamma}_{m}(t)=\frac{1t}{2m}<0.$$
□
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Acknowledgements
The author (SF) was partially supported by JSPS KAKENHI Grant No. 24540146. The author (KY) was also partially supported by JSPS KAKENHI Grant No. 23540208.
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Authors’ contributions
The work presented here was carried out in collaboration between all authors. The study was initiated by SF and the manuscript was written by SF. SF also played the role of the corresponding author. The proof of Lemma 1.1 and the first equality of Eq. (16) in the proof of Lemma 3.3 were given by KY. With the exception of them, the proofs of all results were given by SF. All authors have contributed to, checked and approved the manuscript.
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Furuichi, S., Yanagi, K. Bounds of the logarithmic mean. J Inequal Appl 2013, 535 (2013). https://doi.org/10.1186/1029242X2013535
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Keywords
 logarithmic mean
 matrix mean
 Frobenius norm and inequality