Abstract
The main target of this article is to present several unitarily invariant norm inequalities which are refinements of arithmetic-geometric mean, Heinz and Cauchy-Schwartz inequalities by convexity of some special functions.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
In this sequel, we use the standard notation \(M_{n}, M_{n}^{+}\) and \(M_{n}^{++}\) for the algebra of all \(n\times n\) complex matrices, the cone of positive (or positive semidefinite) matrix and that of strictly positive matrices in \(M_{n}\), respectively. Matrices and their inequalities have attracted researchers working in functional analysis. These inequalities have been studied in different approaches among which unitarily invariant norms inequalities are most popular. Recall that a unitarily invariant norm is a norm \(\Vert \cdot \Vert \) defined on \(M_{n}\) satisfying the property \(\Vert UAV\Vert =\Vert A\Vert \) for all \(A\in M_{n}\) and unitaries \(U,V\in M_{n}\). The absolute value of a matrix \(A = (a_{ij})\) is defined by \(|A| = (A^{*}A)^{1/2}\). The motivation behind this work starts with some crucial inequalities which will be presented as follows.
The classical arithmetic-geometric mean inequality [1] states that for \(A,B\in M_{n}^{+}\) and \(X\in M_{n}\),
Heinz inequality [1] is a refinement of inequality (1) which states that
hold for \(A,B\in M_{n}^{+}, X\in M_{n}\) and \(0\le t\le 1\).
A general form of Cauchy-Schwartz inequality [2] states that for \(A,B\in M_{n}^{+}, X\in M_{n}\) and \(r>0\),
We remark that the above inequalities have been studied deeply in the literature. We refer the reader to [3,4,5] as samples of recent work treating such inequalities and their variants.
Motivated by Bhatia and Bourin [2, 6], here we define two functions f and h for a given unitarily invariant norm \(\Vert \cdot \Vert \),
where \(A,B\in M_{n}^{+}\) and \(X\in M_{n}\). The above functions f and h are convex on [0,1] and attain their minimum at \(t=\frac{1}{2}\). In this article, we utilize convexity of these functions to obtain refinements of arithmetic-geometric mean, Heinz and Cauchy-Schwartz inequalities. The following convex function inequalities are also essential to our results.
Hermite-Hadaward inequality [7] states that for every real-valued convex function g on the interval [a, b], we have
In 2010, EL Farissi [8] refined Hermite-Hadaward inequality as follows
for all \(\lambda \in [0,1]\), where
and
A few years later, Abbas and Mourad [9] got that
The following lemma combining Farissi and Abbas’ results will be essential for our main results. The main results in this paper, Theorems 1, 2 and 3, are obtained by applying some refinements of Hermite-Hadaward inequalities on the convex functions f and h using the same method from Kittaneh [10].
Lemma 1
Let g be a real-valued convex function which is convex on the interval [a,b]. Then for any positive integer n, we have
Recently, Chen, Chen and Gao [11] obtained the following refinements of Hermite-Hadaward inequality.
Lemma 2
Let \(m, n: [a,b]\rightarrow [0,+\infty )\) be convex functions and meet \([m(a)-m(b)]\cdot [n(a)-n(b)]\le 0\). Then for all \(\lambda \in [0,1]\), we have
and
where
and
The organization of this article will be as follows. In the following, we mainly present some unitarily invariant norm inequalities for matrix means which are refinements of arithmetic-geometric mean, Heinz and Cauchy-Schwartz inequalities utilizing Lemmas 1 and 2.
2 Unitarily invariant norm inequalities
Now we are in a position to begin our main results.
Applying Lemma 1 to the convex function h(t) on the interval \([\mu ,1-\mu ]\) when \(0\le \mu <\frac{1}{2}\) and on the interval \([1-\mu ,\mu ]\) when \(\frac{1}{2}<\mu \le 1\), we obtain the following refinement of arithmetic-geometric mean and Heinz inequalities.
Theorem 1
If \(A,B\in M_{n}^{+}\), \(X\in M_{n}\) and \(0\le \mu \le 1\), then for unitarily invariant norm \(\Vert \cdot \Vert \),
hold for any positive integer n.
Proof
Assume that \(A,B\in M_{n}^{+}\), \(X\in M_{n}\) and \(0\le \mu < \frac{1}{2}\), then it follows by Lemma 1 that
which is equivalent to
Hence,
On the other hand, if \(\frac{1}{2}<\mu \le 1\), then it follows by symmetry (i.e., by applying the above inequality (4) to \(1-\mu \)) that
We complete the proof of Theorem 1 by combining the inequalities (4) and (5). \(\square \)
Following the same logic of Theorem 1 and applying Lemma 1 to the function f(t) on the interval \([\mu ,1-\mu ]\) when \(0\le \mu <\frac{1}{2}\) and on the interval \([1-\mu ,\mu ]\) when \(\frac{1}{2}<\mu \le 1\), we have the following refinement of Cauchy-Schwartz inequality.
Theorem 2
If \(A,B\in M_{n}^{+}\), \(X\in M_{n}\) and \(0\le \mu \le 1\), then for unitarily invariant norm \(\Vert \cdot \Vert \),
hold for any positive integer n.
Proof
Assume that \(A,B\in M_{n}^{+}\), \(X\in M_{n}\) and \(0\le \mu < \frac{1}{2}\), then it follows by Lemma 1 that
which is equivalent to
Hence,
On the other hand, if \(\frac{1}{2}<\mu \le 1\), then it follows by symmetry (i.e., by applying the above inequality (6) to \(1-\mu \)) that
We complete the proof of Theorem 2 by combining the inequalities (6) and (7). \(\square \)
Next, for every positive real number r, we consider the function
which is convex on [0,1] and attains its minimum at \(t =\frac{1}{2}\) obtained by Hiai and Zhan [12].
Applying Lemma 1 to the function \(\phi (t)\) on the interval \([\mu ,1-\mu ]\) when \(0\le \mu <\frac{1}{2}\) and on the interval \([1-\mu ,\mu ]\) when \(\frac{1}{2}<\mu \le 1\), then we have the following refinement of general Cauchy-Schwartz inequality.
Theorem 3
If \(A,B\in M_{n}^{+}\), \(X\in M_{n}\) and \(0\le \mu \le 1\), then for unitarily invariant norm \(\Vert \cdot \Vert \),
hold for any positive integer n.
Proof
Assume that \(A,B\in M_{n}^{+}\), \(X\in M_{n}\) and \(0\le \mu < \frac{1}{2}\), then it follows by Lemma 1 that
which is equivalent to
Hence,
On the other hand, if \(\frac{1}{2}<\mu \le 1\), then it follows by symmetry (i.e., by applying the above inequality (8) to \(1-\mu \)) that
We complete the proof of Theorem 3 by combining the inequalities (8) and (9). \(\square \)
In view of the fact that the functions f(t) and h(t) are symmetric, we have
We can have the following result by applying Lemma 2 to function
Corollary 1
For \(0\le \mu \le 1\) and all \(\lambda \in [0,1]\), we have
where
and
Here we remark that \(|f(\mu )-f(1-\mu )|\cdot |\phi (\mu )-\phi (1-\mu )|=0\) and \(|h(\mu )-h(1-\mu )|\cdot |\phi (\mu )-\phi (1-\mu )|=0\) for \(0\le \mu \le 1\). Hence, results similar to Corollary 1 can be obtained by using \(f\cdot \phi \) and \(h\cdot \phi \).
References
Bhatia, R., and C. Davis. 1993. More matrix forms of the arithmetic-geometric mean inequality. SIAM Journal on Matrix Analysis and Applications 14 (1): 132–136.
Bhatia, R. 1997. Matrix analysis. New York: Springer.
Alakhrass, M., and M. Sababheh. 2019. Matrix mixed mean inequalities, Results in Mathematics, 74(1).
Kittaneh, F. 1993. Norm inequalities for fractional powers of positive operators. Letters in Mathematical Physics 27 (4): 279–285.
Sababhehand, M., and M.S. Moslehian. 2017. Advanced refinements of Young and Heinz inequalities. Journal of Number Theory 172: 178–199.
Bourin, J.C. 1999. Some inequalities for norms on matrices and operators. Linear Algebra and its Applications 292 (1–3): 139–154.
Hadamard, J. 1893. Etude sur les proprietes des fonctions entieres et en particulier dune fonction consideree par Riemann. Journal de Mathematiques Pures et Appliquees 58: 171–215.
EL Farissi, A. 2010. Simple proof and refinement of Hermite-Hadamard inequality. Journal of Mathematical Inequalities 4 (3): 365–369.
Abbas, H., and B. Mourad. 2014. A family of refinements of Heinz inequalities of matrices. Journal of Inequalities and Applications 267: 1–7.
Kittaneh, F. 2010. On the convexity of the Heinz mean. Integral Equations Operator Theory 68: 519–527.
Chen, X., Y. Chen, and X. Gao. 2018. A new Hermite-Hadamard Type Inequality, Advances in Intelligent. Systems Research 159: 307–309.
Hiai, F., and X. Zhan. 2002. Inequalities involving unitarily invariant norms and operator monotone functions. Linear Algebra and its Applications 341 (1–3): 151–169.
Acknowledgements
We thank the referee for careful review and valuable comments. This research was supported in part by Key Laboratory of Applied Mathematics of Fujian Province University (Putian University, No. SX201901) and in part by the 2018 Scientific Research Project for Postgraduates of Henan Normal University (Approved Document Number: [2018] 2). This work is funded by National Natural Science Foundation of China with Grant number 11501176.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Ethical approval
This article does not contain any studies with animals performed by any of the authors.
Additional information
Communicated by Samy Ponnusamy.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Zuo, H., Jiang, F. Unitarily invariant norm inequalities for matrix means. J Anal 29, 905–916 (2021). https://doi.org/10.1007/s41478-020-00286-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41478-020-00286-2