Some inequalities for Heinz operator mean

In this paper we obtain some new inequalities for Heinz operator mean.


Introduction
Throughout this paper A, B are positive invertible operators on a complex Hilbert space (H, ·, · ) . We use the following notations for operators and ν ∈ [0, 1] A∇ ν B := (1 − ν) A + νB, the weighted operator arithmetic mean, and the weighted operator geometric mean. When ν = 1 2 we write A∇B and A B for brevity, respectively.
Define the Heinz operator mean by The following interpolatory inequality is obvious for any ν ∈ [0, 1]. The famous Young inequality for scalars says that if a, b > 0 and ν ∈ [0, 1], then with equality if and only if a = b. The inequality (2) is also called ν-weighted arithmetic-geometric mean inequality.
The following inequality provides a refinement and a multiplicative reverse for Young's inequality: The second inequality in (4) is due to Tominaga [12] while the first one is due to Furuichi [4].
The operator version is as follows [4], [12] : For two positive operators A, B and positive real numbers m, m , M, M satisfying either of the following conditions: where h := M m , h := M m and ν ∈ [0, 1] . We observe that, if we write the inequality (5) for 1 − ν and add the obtained inequalities, then we get by division with 2 that that is equivalent to where h := M m , h := M m and ν ∈ [0, 1] . We consider the Kantorovich's constant defined by The function K is decreasing on (0, 1) and increasing on [1, ∞) , K (h) ≥ 1 for any h > 0 and K (h) = K 1 h for any h > 0. The following multiplicative refinement and reverse of Young inequality in terms of Kantorovich's constant holds: The first inequality in (8) was obtained by Zou et al. in [13] while the second by Liao et al. [10].
The operator version is as follows [13], [10]: For two positive operators A, B and positive real numbers m, m , M, M satisfying either of the conditions (i) or (ii) above, we have We observe that, if we write the inequality (9) for 1 − ν and add the obtained inequalities, then we get by division with 2 that that is equivalent to . The inequalities (10) have been obtained in [10] where other bounds in terms of the weighted operator harmonic mean were also given.
Motivated by the above results, we establish in this paper some new inequalities for the Heinz mean. Related inequalities are also provided.

Upper and lower bounds for Heinz mean
We start with the following result that provides a generalization for the inequalities (5) and (9): Theorem 1. Assume that A, B are positive invertible operators and the constants M > m > 0 are such that Then we have the inequalities Proof. From the inequality (4) we have (16) Since, by the properties of Specht's ratio S, we have then by (16) we have for any x ∈ [m, M ] and ν ∈ [0, 1].
Using the functional calculus for the operator X with mI ≤ X ≤ M I we have from (17) that for any ν ∈ [0, 1]. If the condition (11) holds true, then by multiplying in both sides with A −1/2 we get mI ≤ A −1/2 BA −1/2 ≤ M I and by taking X = A −1/2 BA −1/2 in (18) we get Now, if we multiply (19) in both sides with A 1/2 we get the desired result (12). The second part follows in a similar way by utilizing the inequality which follows from (8). The details are omitted.
and by (12) we get and by (12) we get which is equivalent to (20). If we use the inequality (14) for the operators A and B that satisfy either of the conditions (i) or (ii), then we recapture (9).
The following result contains two upper and lower bounds for the Heinz operator mean in terms of the operator arithmetic mean A∇B : Proof. From the inequality (4) we have for ν = 1 for any c, d > 0.
If we take in (29) c = a 1−ν b ν and d = a ν b 1−ν then we get for any a, b > 0 for any ν ∈ [0, 1]. This is an inequality of interest in itself. If we take in (30) a = x and b = 1, then we get for any x > 0. Now, if x ∈ [m, M ] ⊂ (0, ∞) then by (31) we have Then by (32) we have for any x ∈ [m, M ] . If X is an operator with mI ≤ X ≤ M I, then by (33) we have