On further refinements for Young inequalities

In this paper, sharp results on operator Young's inequality are obtained. We first obtain sharp multiplicative refinements and reverses for the operator Young's inequality. Secondly, we give an additive result, which improves a well-known inequality due to Tominaga. We also provide some estimates for $A{{\sharp}_{v}}B-A{{\nabla }_{v}}B$ in which $v\notin \left[ 0,1 \right]$.


Introduction
This note lies in the scope of operator inequalities. We assume that the reader is familiar with the continuous functional calculus and Kubo-Ando theory [6].
It is to be understood throughout the paper that the capital letters present bounded linear operators acting on a Hilbert space H. A is positive (written A ≥ 0) in case Ax, x ≥ 0 for all x ∈ H also an operator A is said to be strictly positive(denoted by A > 0) if A is positive and invertible. If A and B are self-adjoint, we write B ≥ A in case B − A ≥ 0. As usual, by I we denote the identity operator.
The weighted arithmetic mean ∇ v , geometric mean ♯ v , and harmonic mean ! v , for v ∈ [0, 1] and a, b > 0, are defined as follows: If v = 1 2 , we denote the arithmetic, geometric, and harmonic means, respectively, by ∇, ♯ and !, for the simplicity. Like the scalar cases, the operator arithmetic mean, the operator geometric mean, and the operator harmonic mean for A, B > 0 are defined as follows: The celebrated arithmetic-geometric-harmonic-mean inequalities for scalars assert that if a, b > 0, then Generalization of the inequalities (1.1) to operators can be seen as follows: If A, B > 0, then The last inequality above is called the operator Young's inequality. During the past years, several refinements and reverses were given for Young's inequality, see for example [4,5,7].
Zuo et al. showed in [9,Theorem 7] that the following inequality holds: the authors mentioned in [9], the inequality (1.2) improves the following refinement of Young's inequality involving Specht's ratio Another improvement of Young's inequality, is shown in [1, Corollary 1]: We remark that there is no relationship between the constants K(h, 2) r and exp v(1−v) In [3,5] we proved some sharp multiplicative reverses of Young's inequality. In this brief note, as the continuation of our previous works, we establish sharp bounds for the arithmetic, geometric and harmonic mean inequalities. Moreover, we shall show some additive-type refinements and reverses of Young's inequality. We will formulate our new results in a more general setting, namely the sandwich assumption sA ≤ B ≤ tA (0 < s ≤ t). Additionally, we provide

Main Results
In our previous work [3], we gave new sharp inequalities for reverse Young inequalities. In this section, we firstly give new sharp inequalities for Young inequalities, as limited cases in the first inequalities both (i) and (ii) of the following theorem. Proof.
is monotone decreasing for 0 < x ≤ 1 and monotone increasing for x ≥ 1.
Proof. We use again the function f v (x) = (1−v)+vx x v in this proof.

The condition (i) is equivalent to
Note that the second inequalities in both (i) and (ii) of Theorem 2.1 and Corollary 2.1 are special cases in Theorem A of our previous paper [3].

Remark 2.2. It is remarkable that the inequalities
given in the proof of

Consequently,
To see that the constant m∇vM m♯v M in the LHS of (2.2) can not be improved, we consider A = mI and B = MI, then By replacing A, B by A −1 , B −1 , respectively, then the refinement and reverse of noncommutative geometric-harmonic mean inequality can be obtained as follows: Now, we give a new sharp reverse inequality for Young's inequality as an additive-type in the following. Then Therefore, by applying similar arguments as in the proof of Theorem 2.1, we reach the desired inequality (2.4). This completes the proof of theorem.
is the logarithmic mean and the term S (h) refers to the Specht's ratio. Indeed, we have the inequalities which were originally proved in [ Since g v (x) is convex so that we can not obtain a general result on the lower bound for Proof. It follows from the fact that g v (x) is monotone decreasing for 0 < x ≤ 1 and monotone increasing for x ≥ 1.