New Refinement of the Operator Kantorovich Inequality

We focus on the improvement of operator Kantorovich type inequalities. Among the consequences, we improve the main result of the paper [H.R. Moradi, I.H. Gümüş, Z. Heydarbeygi, A glimpse at the operator Kantorovich inequality, Linear Multilinear Algebra, doi:10.1080/ 03081087.2018.1441799].

At the beginning of this paper, we cite the following inequality which is called the operator Kantorovich inequality [1]: where Φ is a normalized positive linear map from B (H) to B (K), (we represent H and K as complex Hilbert spaces throughout the paper) and A is a positive operator with spectrum contained in [m, M] with 0 < m < M.This is a non-commutative analogue of the classical inequality [2], where x ∈ H is a unit vector.
In recent years, various attempts have been made by many authors to improve and generalize the operator Kantorovich inequality.One may see the basic references [3][4][5] and the excellent survey [6] on this topic.In [7], it was shown that (2) The main aim of the present short paper is to improve both inequalities in (2).Actually, we prove that where In what follows, an operator means a bounded linear one acting on a complex Hilbert space H.As customary, we reserve m, M for scalars and I for the identity operator.A self-adjoint operator A is said to be positive if Ax, x ≥ 0 holds for all x ∈ H.A linear map Φ is positive if Φ (A) ≥ 0 whenever A ≥ 0. It is said to be normalized if Φ (I) = I.We denote by σ (A) the spectrum of the operator A.

Main Results
Before we present the proof of our theorems, we begin with a general observation.We say that a non-negative function f on [0, ∞) is geometrically convex [8] when for all a, b > 0 and v ∈ [0, 1].Equivalently, a function f is geometrically convex if and only if the associated function F (y) = log ( f (e y )) is convex.
Example 1 ([9] Example 2.12).Given real numbers c i ≥ 0 and p Kittaneh and Manasrah [10] Theorem 2.1 obtained a refinement of the weighted arithmetic-geometric mean inequality as follows: where r = min {v, where the first inequality follows from the inequality (1 + vb and the fact that f is decreasing function, in the second inequality we used (4), the third inequality is obvious by (3), and the fourth inequality again follows from (4) by interchanging a by f (a) and b by f (b).
Of course, each decreasing geometrically convex function is also convex.However, the converse does not hold in general.
The inequality (5 In order to establish our promised refinement of the operator Kantorovich inequality, we also use the well-known monotonicity principle for bounded self-adjoint operators on Hilbert space (see, e.g., [6] provided that f and g are real-valued continuous functions.Under the same assumptions, h (t) = |t| implies h (A) = |A|.Now, we are in a position to state and prove our main results.We remark that the following theorem can be regarded as an extension of [5] Remark 4.14 to the context of geometrical convex functions.Theorem 1.Let A ∈ B (H) be a self-adjoint operator with σ (A) ⊆ [m, M] for some scalars m, M with 0 < m < M and Φ be a normalized positive linear map from B (H) to B (K).If f is strictly positive decreasing geometrically convex function, then Proof.On account of the assumptions, from parts of (6), we have where Note that inequality (8) holds for all t ∈ [m, M].On the other hand, σ (A) ⊆ [m, M], which, by virtue of monotonicity principle (7) for operator functions, yields the series of inequalities It follows from the linearity and the positivity of the map Φ that Now, by using [5] Corollary 4.12 we get This completes the proof.
As discussed extensively in [6] Cahpter 2, for f (t) = t p , we have Now, the following fact can be easily deduced from Theorem 1 and Example 1.

Corollary 1.
Let A ∈ B (H) be a positive operator with σ (A) ⊆ [m, M] for some scalars m, M with 0 < m < M and Φ be a normalized positive linear map from B (H) to B (K).Then for any p < 0, where In particular, We note that K (m, M, −1) = (M+m) 2 4Mm is the original Kantorovich constant.
Theorem 2. Let all the assumptions of Theorem 1 hold.Then Proof.By applying a standard functional calculus for the operator Φ(A) such that mI ≤ Φ (A) ≤ MI, we get from (8) We thus have where at the last step we used the basic inequality [5] Corollary 4.12.
Hence, the proof is complete.
As a corollary of Theorem 2 we have: Corollary 2. Let all the assumptions of Corollary 1 hold.Then for any p < 0 Remark 1.Notice that the inequalities in Corollary 2 are stronger than the inequalities obtained in [11] Corollary 2.1.
Recall that if f is operator convex, the solidarities [12] or the perspective [13] of f is defined by Using a series of inequalities (6) we have the upper bounds of the perspective for non-negative decreasing geometrically convex function (not necessary operator convex f ).We use the same symbol P f (A | B) for a simplicity.Proposition 1.Let A, B > 0 with mA ≤ B ≤ MA for some scalars 0 < m < M. For a non-negative decreasing geometrically convex function f , we have