New versions of re ﬁ nements and reverses of Young - type inequalities with the Kantorovich constant

: Recently, some Young - type inequalities have been promoted. The purpose of this article is to give further re ﬁ nements and reverses to them with Kantorovich constants. Simultaneously, according to the scalar result, we have obtained some corresponding operator inequalities and matrix versions, including Hilbert - Schmidt norm, unitary invariant norm, and trace norm can be regarded as Scalar inequality.

and for all the unitary matrices ( ) , the Hilbert-Schmidt (or Frobenius) norm and the trace norm of A are defined by respectively, where ( ) are the singular values of A, that is, the eigenvalues of the positive matrix | | ( ) , arranged in decreasing order and repeated according to multiplicity.Moreover, it is well known that ∥ ∥ . 2 is unitarily invariant.Furthermore, let ( ) be the * C -algebra of all bounded linear operators on a complex separable Hilbert space .I stands for the identity operator.
( ) denotes the cone of all positive invertible operators on .As a matter of convenience, we use the following notations to define the ν-weighted arithmetic-mean (AM), geometric-mean (GM), and harmonic-mean (HM) for scalars and operators: ) ) where > a b , 0, ( ) , we write ∇ a b, ∇ A B, ♯ A B, and ! a b for brevity, respectively.
In [13,14], Kittaneh and Manasrah researched Young's inequality and obtained the following results: where . The left inequality of (2) can be regarded as a refinement of Young's inequality and the right one can be regarded as a reverse of Young's inequality.
In [25], Zhao and Wu deepened inequality (2) and showed another refinement of Young's inequality as follows: where , and Zhao and Wu also obtained a more precise form of the reverse Young's inequality as follows: where , and In a recent work, Hu [8] introduced a refinement of Young-type inequalities as: (5) Later, based on the result of Hu [8], Nasiri et al. [16] have given some refinements of (5).These inequalities can be written as: where where . Later, Yang and Li [24] established some refinements and reverses of (6) and (7).These inequalities can be written as: , then where , and The Kantorovich's constant defined by ( ) and monotone decreasing on ( ] 0, 1 .
The following multiplicative refinement and reverse of Young's inequality in terms of Kantorovich's constant holds: where , and = h b a .The second inequality in (12) is due to Liao et al. [11], whereas the first one is due to Zou et al. [27].In [23], Zhao and Wu made a further study about inequality (2) with the Kantorovich constant and gave the following results: where where . Let us take a closer look at [11] where Liao et al. made a reverse refinement for Young's inequality as follows: where . In [10], Liao and Wu further deepened the results of inequality (14) and obtained the following results: , then New versions of refinements and reverses  3 , then where , and Recently, Zhao and Wu [26] obtained the refinements and reverses of Young's inequality and improved inequalities (2) in the following forms: If where 1, 2 4 , and  .Following [11,27], the operator version is as follows : For two positive operators A B , and positive real numbers ′ ′ m m M M , , , satisfying either of the following conditions: , and . In [15], inequalities (6) and (7) for the Hilbert-Schmidt norm form with Kantorovich constant were obtained by Nasiri and Shakoori, that is, if where where For a detailed study of these and associated norm inequalities along with their history of origin, refinements, and applications, one may refer to [1][2][3][4][5]11,[18][19][20][21][22].
One of the goals of this article is to obtain new refinements and reverses of inequalities ( 8)- (11) with the Kantorovich constant.
The article is organized as follows.In Section 2, a new scalar version of Young's inequality and its reverse with the Kantorovich constant are first given.In Section 3, we obtain the corresponding new operator inequalities on the Hilbert space.Finally, in Section 4, we mainly show the matrix version of inequalities for the Hilbert-Schmidt norm, unitarily invariant norm, and related trace version based on the result of part one.

New progress of Young's and its reverse inequalities
In this section, we mainly present the improved scalar Young and its reverse inequalities related to the Kantorovich constant.(29) 1 , and New versions of refinements and reverses  5 (30) 2, 3 4 , and , then (31) 3, 4 4 , and Proof.The process of the proof of inequalities (30) and (31) is similar to that of inequalities (28) and (29), so we only need to prove inequalities (28) and (29).We first consider the case . By the left side of inequality (21), we have and so, ( . By the right side of inequality (21), we have and so, This completes the proof of inequality (28).
Consider the case . By the left side of inequality (22), we have and so, .
Now, by the right side of inequality (22), we have and so The proof is complete.□ Remark 2.2.By the property of the Kantorovich constant, the inequalities in Theorem 2.1 are the improved results of [24].
Proof.The process of the proof of the second inequality is similar to that of the first one, we only need to investigate the first one.By inequality (17), we have and so Now, by inequality (19), we have and hence So, the proof is complete.□ Next we are going to deduce another form of reverse ratio Young's inequality by virtue of inequality (16).
Theorem 2.4.Suppose that > a b , 0, and where , then where Proof.The process of the proof of the second inequality is similar to that of the first one, we only need to investigate the first one.By inequality (16), we have and so, This completes the proof.□ 3 New operator versions of Young-type inequalities In the section, we give some more excellent versions of Young-type operator inequalities and their reverse by the monotonic property of operator functions.First, we present the monotonic property of operator function, which is the basis of the following discussion.
, then: (36) New versions of refinements and reverses  9 (37) 1 , and , then (39) 3, 4 4 , and Proof.We only need to investigate inequality (36) due to the similarity of the process of proof., we have

I hI I X h I I
. By Lemma 3.1, we have Since the Kantorovich constant is an increasing function on , then and Since the Kantorovich constant is a decreasing function on ( ) 0, 1 , then (41) New versions of refinements and reverses  11 , then: , (42) , then where Proof.The process of the proof is analogous to that of Theorem 3.2, so we omit it here.
, where B is a positive operator, C is invertible, = * A C C and I is the identity operator and where ( ) Proof.The process of the proof of the second inequality is similar to that of the first one, we only need to investigate the first one.By inequality (34), we have , under the first condition, we have .
Since the Kantorovich constant is an increasing function on , by (46), we have It can be deduced from inequality (47) and Lemma 3.1 that Likewise, under the second condition, we also have that < and then (49) Since the Kantorovich constant is a decreasing function on ( ) 0, 1 , by (49), we have It can be deduced from inequality (50) and Lemma 3.1 that It is striking that we can obtain the same inequality (48) under the first condition or the second condition one.Next, multiplying inequality (48) or (51) by * C on the left and C on the right, then we can deduce the required inequality (44).The proof is complete.□ 4 New matrix versions of Young's inequalities for the Hilbert-Schmidt norm Only some interesting matrix versions of Theorem 2.1 for Hilbert-Schmidt norm, unitarily invariant norm, trace norm, and trace are discussed in this section.We need the following lemmas to do this.It is, though, worth noting.For unitarily invariant norms, the first lemma is a Heinz-Kato form inequality.
In particular, s AB s A s B .Cauchy-Schwarz's inequality.Let ≥ a b , 0 Now, we first establish a matrix version of Theorem 2.1 for the Hilbert-Schmidt norm, the proof is based on the spectrum theorem.where

, then
New versions of refinements and reverses  15 , then  Proof.Since every positive semi-definite is unitarily diagonalizable, it follows by spectral theorem that there are unitary matrices , and , then by inequality (28) and the unitary invariance of the Hilbert-Schmidt norm, we have   So we completed the proof of (52).The proof of the remaining cases are similar to the proof of case (a), so we omit it.□ Proof.We shall prove the first inequality and leave the others to the reader because the proof is similar to each other.
then Zhao and Wu made a reverse refinement of the second inequality in(2)

1 .
Then we have the following:

New versions of refinements and reverses  17 4 (
using Lemmas 4.2 and 4.3, Cauchy-Schwarz inequality, and the left inequality of (28), we have ,