Reverse and improved inequalities for operator monotone functions

. In this paper we provide several reﬁnements and reverse operator inequalities for operator monotone functions in Hilbert spaces. We also obtain reﬁnements and a reverse of L¨owner–Heinz celebrated inequality that holds in the case of power function.


Introduction. Consider a complex Hilbert space (H, •, •
).An operator T is said to be positive (denoted by T ≥ 0) if T x, x ≥ 0 for all x ∈ H and also an operator T is said to be strictly positive (denoted by T > 0) if T is positive and invertible.A real valued continuous function f (t) on [0, ∞) is said to be operator monotone if f (A) ≥ f (B) holds for any A ≥ B ≥ 0, which is defined as A − B ≥ 0.
In 1934, K. Löwner [6] had given a definitive characterization of operator monotone functions as follows, see for instance [1, p. 144-145]: where b ≥ 0 and m is a positive measure on [0, ∞) such that We recall the important fact proved by Löwner and Heinz which states that the power function f : [0, ∞) → R, f (t) = t α is an operator monotone function for any α ∈ [0, 1], see [5].
Consider the family of functions defined on (0, ∞) by We also have the functions of interest: In [2], the authors showed that f p is operator monotone for 1 ≤ p ≤ 2. In the same category, we observe that the function is an operator monotone function for p ∈ (0, 1], see [3].
It is well known that the logarithmic function ln is operator monotone and in [3], the author proved that the functions Let A and B be strictly positive operators on a Hilbert space H such that B − A ≥ m1 H > 0. In 2015, T. Furuta [4] obtained the following result for any non-constant operator monotone function f on [0, ∞): The inequality between the first and third term in (1.3) was obtained earlier by H. Zuo and G. Duan in [9].
By taking f (t) = t r , r ∈ (0, 1] in (1.3), Furuta obtained the following refinement of the celebrated Löwner-Heinz inequality With the same assumptions for A and B, we have the logarithmic inequality [4]: Notice that the inequalities between the first and third terms in (1.4) and (1.5) were obtained earlier by M. S. Moslehian and H. Najafi in [8].
Motivated by the above results, we show in this paper that if Some examples of interest, including a refinement and a reverse of the Löwner-Heinz inequality, are also provided.

Main Results.
We have: given by representation (1.1).Let A ≥ 0 and assume that there exist positive numbers d > c > 0 such that then f can be written as in the equation (1.1) and for A, B ≥ 0 we have the representation Observe that for s > 0, Therefore, (2.3) becomes (see also [4]) The function g (t) = −t −1 is operator monotone on (0, ∞), operator Gâteaux differentiable and the Gâteaux derivative is given by for T, S > 0.
Consider the continuous function g defined on an interval I for which the corresponding operator function is Gâteaux differentiable and for selfadjoint operators C, D with spectra in I we consider the auxiliary function defined on then, by the properties of the Bochner integral, we have If we write this equality for the function g (t) = −t −1 and C, D > 0, then we get the representation By the representation (2.4), we derive the following identity of interest (2.9) for A, B ≥ 0.
From the first inequality in (2.2) we get and since d1 H − (B − A) ≥ 0 and b ≥ 0, From the second inequality in (2.2) we have Therefore we have the following result which does not contain the value b: A ≥ 0 and that there exist positive numbers d > c > 0 such that the condition (2.1) is satisfied.Then Remark 1.If we take f (t) = t r , r ∈ (0, 1], in (2.14), then we get provided that the condition (2.1) is satisfied and A ≥ 0.
Let ε > 0. Consider the function f ε : [0, ∞) → R, f ε (t) = ln (ε + t).This function is operator monotone on [0, ∞) and by the second inequality in (2.14), we get (2.16) By taking the limit over ε → 0+ in (2.16), we get (2.17) It is well known that if P ≥ 0, then x, T −1 x = T x, x T −1 , which implies the following operator inequality The function f (t) = ln (t + 1) is also operator monotone on [0, ∞), so by (2.19) we have The interested reader may state other similar inequalities by employing the operator monotone functions presented in Introduction.We omit the details.