Bounds of the logarithmic mean

We give tight bounds for logarithmic mean. We also give new Frobenius norm inequalities for two positive semidefinite matrices. In addition, we give some matrix inequalities on matrix power mean.


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 [3], the following relations were shown.
We now have the following lemma.
The first inequality of the inequalities (3) refines the inequality which is known as classical Pólya inequality [2,4]. Throughout this paper we use the notation M (n, C) as the set of all n × n matrices on the complex field C. We also use the notation M + (n, C) as the set of all n × n positive semidefinite matrices. Here A ∈ M + (n, C) means we have φ|A|φ ≥ 0 for any vector |φ ∈ C n . For A ∈ M (n, C), the Frobenius norm (Hilbert-Schmidt norm) · F is defined by In the paper [2], the following norm inequality was shown.
) For A, B ∈ M + (n, C), X ∈ M (n, C) and Frobenius norm · F , we have From Lemma 1.1, we have the following proposition.
To the first author's best knowledge, the first inequality in Proposition 1.3 was suggested in [5]. This proposition can be proven by the 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 the inequality (6) shown in [2].

Lower bound of logarithmic mean
The following inequalities were given in [1]. Hiai and Kosaki gave the norm inequalities for Hilbert space operators in [1]. See also [6,7]. Here we give them as a matrix setting to unify the description of this paper.
Frobenius norm is one of unitarily invariant norms. We give the refinement of the lower bound of the above first inequality for Frobenius norm. That is, we have the following inequalities.
Theorem 2.2 For A, B ∈ M + (n, C), X ∈ M (n, C), m ≥ 1 and Frobenius norm · 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 ≥ 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 = 1 and v = 0. In addition, for the case that u = w, the equality holds. Thus we may assume w > u and v ≥ 1.
Then the lemma can be proven by the following way. x Proof : For the case x = 1, the equality holds. So we prove this lemma for x = 1. If m is an odd number, then we have m = 2⌊m/2⌋ + 1. Since we then have m + 1 = 2(⌊m/2⌋ + 1) and If we putk k = ⌊m/2⌋ + 1, then the above means (2k k − 1)(m + 1) = 2k k m. Then the difference of thek k-th term of the both sides in the inequality (9) is equal to 0. For the case that m is an even number, it never happens that the difference of thek k-th term of the both sides in the inequality (9) is equal to 0. Therefore we have m k=1 where a l = (2l − 1)(m + 1), b l = m − (2l − 1) and c l = 2{m − (l − 1)}m, for l = 1, 2, · · · , ⌊m/2⌋.
Thus the proof of this lemma was completed.
We then have the following lemma.
Proof : The second inequality follows by the inequality (11). We use the famous inequality which implies the first inequality. The third inequality can be proven by the use of the arithmetic mean -geometric mean inequality. Thus the proof of this lemma was completed.
We give some basic properties of the right hand side of the inequality (13) in Appendix.

Upper bound of logarithmic mean
In the paper [1], 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.
We also give an improved upper bound of the logarithmic mean on the above Theorem 3.1, only for the Frobenius norm. Namely, we can prove the following inequalities by the 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 ∈ M + (n, C), X ∈ M (n, C), m ≥ 2 and Frobenius norm · F , we have To prove Theorem 3.2 we need to prove the following lemmas.
Proof : To prove the first inequality, we have only to prove the following inequality The inequality (19) can be proven by putting x ≡ t 1/m > 0 in a famous inequality (x−1)/log x ≤ (x + 1)/2 for x > 0.
To prove the second inequality of the inequalities (18), it is sufficient to prove the inequality (15) which holds from Lemma 3.3. We obtain actually the second inequality of the inequalities (18) by putting t = x m(m−1) > 0 in the inequality (15), and then putting t = a/b.
To prove the third inequality of the inequalities (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 to he inequality (20). Then we have Therefore we have only to prove the inequality which is equivalent to the inequality We give some basic properties of the right hand side of the inequality (19) in Appendix.

Matrix inequalities on geometric mean
We give the proof of Proposition 4.3. Proposition 4.1 and Proposition 4.2 are also proven by the 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 operator mean m, it is well-known [8] that f m (x) ≤ f n (x) holds for x > 0 if and only if AmB ≤ AnB holds for all positive operators A and B. However we give an elementary proof for the convenience of the readers.
Closing this section, we give another matrix inequalities by the use of the another lower bound of the logarithmic mean. As an 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 to write them. From the inequalities (21), we have The inequalities (22) imply the following result by the similar way to the proof of Proposition 4.3.

Comments
Proposition 5.2 given in 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 α m (t) of the logarithmic mean L(t, 1) is tighter than the lower bound √ t given by T.-P.Lin in [3], for m ≥ 1. However, it may be difficult problem to find the minimum m ∈ N such that β m (t) ≤ ((t 1/3 + 1)/2) 3 for any t > 0. The right hand side of the above inequality is the upper bound given by T.-P.Lin in [3]. (See the inequalities (2).) Since g ′ m (r) = (2m + 1)r m−1 (r m+1 − (m + 1)r + m) > 0, we have g m (r) > g m (1) = 0, which implies m(r m + 1)(r m+1 − 1) < (m + 1)(r m+1 + 1)(r m − 1).
As standard bounds of Riemann sum for the integral Then we have the following relations.