Some results of Heron mean and Young’s inequalities

In this paper, we will show some improvements of Heron mean and the refinements of Young’s inequalities for operators and matrices with a different method based on others’ results.


Introduction
For two positive numbers a, b and v ∈ [0, 1], the quantity is called Heron mean. And the inequality is called Young's inequality. Even though these inequalities look very simple, they have attracted many researchers in this field, where adding a positive term to refine the inequalities is possible. Heron mean is the interpolation between arithmetic and geometric means for a, b ≥ 0 and v ∈ [0, 1]. We can see papers [1,2], and [3] for some new results about Heron mean and arithmetic-geometric mean.
The first refinements of Young's inequality is the squared version proved in [4] a v b 1-v 2 + min{v, 1 -v} 2 (ab) 2 Later, the authors in [5] obtained the other interesting refinement (

1.4)
A common fact about refinements (1.2) and (1.3) is having one refining term.
In this paper, our main results are to give refinements of Heron mean for scalars and matrices in Sect. 2; and in Sect. 3, in a different way, to get an operator version of (1.5), which is the refinement of (1.4). Besides, in the same section, the refinements of Young's inequalities for the Hilbert-Schmidt norm will be presented using the same technology as in Sect. 2.
For our convenience, we firstly give some denotations.
It is well known that the Hilbert-Schmidt norm is unitarily invariant in the sense that |UAV | = |A | for all unitary matrices U, V ∈ M n . What is more, we define denoted by A∇B and A B, respectively, when v = 1 2 .

Refinements of Heron mean
Heron mean is defined by It is easy to see that F v (a, b) is an increasing function in v on [0, 1] and Our purpose of this section is to give refinements of Heron mean for a scalar and some other auxiliary results.
Proof Firstly, Next, we also have 3) and (2.4) are refinements of (2.2). With Theorem 2.1 in hand, we will give refinements of Heron mean for operators by the monotonicity property of operator functions.

Lemma 2.2 Let X ∈ B(H) be self-adjoint, and let f and g be continuous real functions such
For more details about this property, the reader is referred to [7].
3) and expand the summand to get Note that the operator X = A -1 2 BA -1 2 has a positive spectrum, and by Lemma 2.2 and (2.7) we have Finally, multiplying inequality (2.8) by A 1 2 on the left-and right-hand sides, we can get which is equivalent to (2.5).
Using the same technique in (2.4), we can get (2.6). So we completed the proof.
Next, we will present the refinements of Heron mean for the Hilbert-Schmidt norm.
Now, by (2.10) and the unitary invariance of the Hilbert-Schmidt norm, we have So we finished the proof.

Refinements of Young's inequalities
It is well known that with equality if and only if a = b is called Young's inequality. An operator version of (2.11) in [7] says that for A, B ∈ B + (H) and v ∈ [0, 1]. Kittaneh and Manasrah [5] gave a different type of improvement of Young's matrix inequalities: for A, B ∈ B + (H), v ∈ [0, 1], r = min{v, 1 -v}, and s = max{v, 1 -v}.
Here, we give the first inequalities' refinements of (2.13). Before that, we need a lemma.
Proof For 0 ≤ ν ≤ 1 2 , then we have 0 ≤ 2ν ≤ 1. Substituting B by A B and ν by 2ν in the first inequality (2.13), we have (2.16) By computing directly with Lemma 2.5, then we have Exchanging A for B and ν for 1ν in (2.17) for 1 2 ≤ ν ≤ 1, we get So we completed the proof.
Proof For 0 ≤ ν ≤ 1 4 , then r = ν, r 1 = 2ν. So (2.19) is equivalent to that is, So we only need to prove   Here, we should remind the readers that the reverse of Theorem 2.6 is stronger than (2.13) and only holds for 0 ≤ ν ≤ 1 4 and 3 4 ≤ ν ≤ 1 in Zhao and Li [8]. We also can see Zhao and Wu [6] for a different method to get Theorem 2.6.

Conclusion
In order to better estimate the Heron mean, a refinement inequality about the classical interpolation between arithmetic mean and geometric mean by Heron mean is obtained, which is also applicable to establishing the inequalities for operators and matrices. Next an operator version refinement inequality about Young's inequality is also established, which is a generalization on the results obtained previously by Kittaneh and Manasrah [5]. It is worth noting that the inequality mentioned can also give the refinement inequality about Young's inequality, which was presented by Zhao and Wu [6].