Enhanced Young-type inequalities utilizing Kantorovich approach for semide ﬁ nite matrices

: This article introduces new Young-type inequalities, leveraging the Kantorovich constant, by re ﬁ ning the original inequality. In addition, we present a range of norm-based inequalities applicable to positive semide ﬁ nite matrices, such as the Hilbert-Schmidt norm and the trace norm. The importance of these results lies in their dual signi ﬁ cance: they hold inherent value on their own


Introduction
Consider two positive numbers, denoted as a and b.Following the widely recognized Young inequality, as formulated by where ν ranges from 0 to 1 inclusively.To facilitate the discussion, we introduce the concepts of weighted arithmetic and geometric means as follows: and by employing a weighted arithmetic-geometric means inequality, we can express the Young inequality as follows: In scholarly discussions, this inequality has been garnering increasingly more attention.An advancement in the Young inequality was proposed by Kittaneh and Manasrah in their work [1], which can be expressed as follows: where the parameter r 0 is defined as the minimum of ν and − ν 1 .The Young inequality underwent further enhancement through the subsequent extension by Hirzallah and Kittaneh, as detailed in their work [2], yielding the ensuing inequality (1.5) where r 0 is determined as the minimum of ν and − ν 1 .
The subsequent multiplicative refinement and reverse of the Young inequality, formulated with respect to Kantorovich constant, can be expressed as follows: , 2 ; here, it is essential to note that a and b are both positive, ν lies within the interval [ ] 0, 1 , and we define , and = h b a .The second inequality within equation (1.6) is credited to Liao et al. [8], while the first inequality is attributed to Zou et al. [7].
In reference to [9], the authors have achieved an enhanced version of the Young inequality and its converse, which are described as follows: First, the multiplicative refinement and reverse of the Young inequality, incorporating Kantorovich constant, are stated as follows: where the parameters h, r, R, and ′ r are defined as follows: h is the ratio b a , r is the minimum of ν and − ν 1 , R is the maximum of ν and − ν 1 , and ′ r is the minimum of r 2 and − r 1 2 .In addition, another variation of the reverse Young inequality, utilizing Kantorovich constant, is elucidated in [8], employing the same notation as mentioned above: where ′ R is calculated as the maximum of r 2 and − r 1 2 .Furthermore, Rashid and Bani-Ahmad [5] have recently refined the inequalities (1.7) and (1.8) with the following results: (i) When ν falls within the range ⎡ ⎣ ⎤ ⎦ 0, 1 2 , the inequalities are given as follows: (1.10) where = h a b , r is the minimum of ν 2 and − ν 1 2 , r 0 is the minimum of r 2 and − r 1 2 , and r 1 is the minimum of r 2 0 and − r , the inequalities are expressed as follows: The Young inequality for two real numbers, serving as a weighted extension of the generalized arithmeticgeometric mean inequality, constitutes a fundamental connection between two nonnegative real values, and we acknowledge its significance.For further elaboration on this type of inequality, the author suggests referring to [1,4,[10][11][12][13][14] and the related references.
This study focuses on enhancing and reversing the inequalities presented in equations (1.10) and (1.11).In addition, we offer several matrix inequalities employing the Hilbert-Schmidt norm and trace norm as practical applications of our methodology.
This article comprehensively addresses all of these topics, and its structure is organized as follows: in Section 2, we derive several supplementary results, which serve as refinements and reversals of the inequalities (1.4)-(1.9),(1.10), and (1.11); in Section 3, we establish the matrix counterparts of the inequalities (2.1)-(2.7)for both the Hilbert-Schmidt norm and the trace norm.We accomplish this task by relying on Lemmas 3.1, 3.2, 3.3, and the Cauchy-Schwarz inequality.
2 Scalar-type inequalities using mean By applying the inequality (1.7) for the relation (2.2), it follows that Hence, the inequality (2.1) follows.
Proof.By applying the inequality (1.8) for the relation (2.2), it follows that So, we obtain the inequality (2.3). (2.4) 2 1 , and Proof.The proof of the second inequality follows by applying again Lemma 2.1, but to a, ( ) We only need to investigate the first one.Applying Lemma 2.1 to νa b , and parameter ν 2 , we have So, the result.3 Norm and trace-type inequalities using matrix means In the subsequent text, we denote the space of all complex matrices of size × n n as ( ) , the Hilbert-Schmidt (or Frobenius) norm and the trace norm of A are de- fined as follows: where represent the singular values of A, which are essentially the eigenvalues of the positive matrix . These singular values are arranged in decreasing order and repeated according to their multiplicity.It is a well-known fact that ∥ ∥ . 2 is invariant under unitary transformations (as elaborated in [16] and [17]).
In the realm of matrices within the space ( ) M n , particularly focusing on matrices denoted as A and B that fall under the category of positive semidefinite matrices, Rashid and Bani-Ahmad, as presented in [5, Theorem 4.5], have innovatively introduced a matrix-based counterpart to the inequalities (1.10) and (1.11).
In this section, we will provide a concise overview of the Hilbert-Schmidt norm, unitarily invariant norm, trace norm, and trace as they pertain to the intriguing matrix formulations found in Theorem 2.3.To facilitate this discussion, we will make use of the following Lemmas.It is important to emphasize that the initial lemma, concerning unitarily invariant norms, takes the form of a Heinz-Kato inequality.
In particular, Enhanced Young-type inequalities  5 Cauchy-Schwarz's inequality.Let ≥ a b , 0 We will now establish a matrix counterpart of Theorem 2.3 specifically designed for the Hilbert-Schmidt norm.The proof of this matrix version relies on the spectrum theorem. , and I is the identity matrix, and Then, we have the following: 2 1 , and Proof.Since every positive semidefinite is unitarily diagonalizable, it follows by spectral theorem that there are unitary matrices , where , and The following system of equations describes the relationship between the matrices A X , , and B in terms of the elements y ij of the matrix Y : If ≤ < ≤ ν κ 0 1 2 , then by inequality (2.4) and the unitary invariance of the Hilbert-Schmidt norm, we have and the property of the Kantorovich constant, we have where , then where  where and the property of the Kantorovich constant, we have   This study has introduced novel Young-type inequalities, capitalizing on the Kantorovich constant, by enhancing the original inequality.Furthermore, a collection of norm-based inequalities has been presented, specifically designed for positive semidefinite matrices, encompassing norms like the Hilbert-Schmidt norm and the trace norm.The significance of these findings is twofold: they possess intrinsic value in their own right and serve as extensions and advancements of numerous established results in the existing literature.As for future work, there is a promising avenue for further exploration in this area.Future research could focus on applying these inequalities in various mathematical and scientific contexts, potentially leading to new insights and applications.In addition, the refinement and extension of these inequalities to more general matrix settings or other mathematical structures could also be an intriguing direction for future investigations.

,
and r 1 is the minimum of r 2 0 and − r 1 2 0 .
The following results for unitarily invariant norm trace are established by Lemma 3.1.