## Banach Journal of Mathematical Analysis

### Interpolating inequalities for functions of positive semidefinite matrices

#### Abstract

Let $A$, $B$ be positive semidefinite $n\times n$ matrices, and let $\alpha\in(0,1)$. We show that if $f$ is an increasing submultiplicative function on $[0,\infty)$ with $f(0)=0$ such that $f(t)$ and $f^{2}(t^{1/2})$ are convex, then $\begin{eqnarray*}\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert f(\llvert AB\rrvert )\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert^{2}&\leq&f^{4}(\frac{1}{(4\alpha(1-\alpha))^{1/4}})(\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert(\alpha f(A)+(1-\alpha )f(B))^{2}\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert\\&&{}\times \big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert((1-\alpha)f(A)+\alpha f(B))^{2}\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert)\end{eqnarray*}$ for every unitarily invariant norm. Moreover, if $\alpha\in{}[0,1]$ and $X$ is an $n\times n$ matrix with $X\neq0$, then $\begin{eqnarray*}&&\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert f(\llvert AXB\rrvert )\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert^{2}\\&&\quad \leq\frac{f(\Vert X\Vert)}{\Vert X\Vert}\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert\alpha f^{2}(A)X+(1-\alpha)Xf^{2}(B)\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert \big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert(1-\alpha)f^{2}(A)X+\alpha Xf^{2}(B)\big\vert\hspace*{-0.8pt}\big\vert\hspace*{-0.8pt}\big\vert\end{eqnarray*}$ for every unitarily invariant norm. These inequalities present generalizations of recent results of Zou and Jiang and of Audenaert.

#### Article information

Source
Banach J. Math. Anal., Volume 12, Number 4 (2018), 955-969.

Dates
Accepted: 12 March 2018
First available in Project Euclid: 10 July 2018

https://projecteuclid.org/euclid.bjma/1531209674

Digital Object Identifier
doi:10.1215/17358787-2018-0008

Mathematical Reviews number (MathSciNet)
MR3858756

Zentralblatt MATH identifier
06946298

#### Citation

Al-Natoor, Ahmad; Hirzallah, Omar; Kittaneh, Fuad. Interpolating inequalities for functions of positive semidefinite matrices. Banach J. Math. Anal. 12 (2018), no. 4, 955--969. doi:10.1215/17358787-2018-0008. https://projecteuclid.org/euclid.bjma/1531209674

#### References

• [1] M. Alakhrass, A note on Audenaert interpolating inequality, preprint, http://www.tandfonline.com/doi/full/10.1080/03081087.2017.1376614 (accessed 25 May 2018).
• [2] H. Albadawi, Singular value and arithmetic-geometric mean inequalities for operators, Ann. Funct. Anal. 3 (2012), no. 1, 10–18.
• [3] M. Al-khlyleh and F. Kittaneh, Interpolating inequalities related to a recent result of Audenaert, Linear Multilinear Algebra 65 (2017), no. 5, 922–929.
• [4] K. M. R. Audenaert, Interpolating between the arithmetic-geometric mean and Cauchy-Schwarz matrix norm inequalities, Oper. Matrices 9 (2015), no. 2, 475–479.
• [5] M. Bakherad, R. Lashkaripour, and M. Hajmohamadi, Extensions of interpolation between the arithmetic-geometric mean inequality for matrices, J. Inequal. Appl. 2017, no. 209.
• [6] R. Bhatia, Matrix Analysis, Grad. Texts in Math. 169, Springer, New York, 1997.
• [7] R. Bhatia and C. Davis, More matrix forms of the arithmetic-geometric mean inequality, SIAM J. Matrix Anal. Appl. 14 (1993), no. 1, 132–136.
• [8] R. Bhatia and C. Davis, A Cauchy-Schwarz inequality for operators with applications, Linear Algebra Appl. 223/224 (1995), 119–129.
• [9] R. Bhatia and F. Kittaneh, On the singular values of a product of operators, SIAM J. Matrix Anal. Appl. 11 (1990), no. 2, 272–277.
• [10] R. Bhatia and F. Kittaneh, Notes on matrix arithmetic–geometric mean inequalities, Linear Algebra Appl. 308 (2000), no. 1–3, 203–211.
• [11] R. Bhatia and F. Kittaneh, The matrix arithmetic-geometric mean inequality revisited, Linear Algebra Appl. 428 (2008), no. 8–9, 2177–2191.
• [12] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr. 18, Amer. Math. Soc., Providence, 1969.
• [13] Y. Han and J. Shao, Notes on two recent results of Audenaert, Electron. J. Linear Algebra 31 (2016), 147–155.
• [14] F. Hiai and X. Zhan, Inequalities involving unitarily invariant norms and operator monotone functions, Linear Algebra Appl. 341 (2002), no. 1–3, 151–169.
• [15] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge, 1991.
• [16] R. A. Horn and R. Mathias, Cauchy-Schwarz inequalities associated with positive semidefinite matrices, Linear Algebra Appl. 142 (1990), 63–82.
• [17] Y. Kapil, C. Conde, M. S. Moslehian, M. Singh, and M. Sababheh, Norm inequalities related to the Heron and Heinz means, Mediterr. J. Math. 14 (2017), no. 5, art ID 213.
• [18] F. Kittaneh, A note on the arithmetic-geometric mean inequality for matrices, Linear Algebra Appl. 171 (1992), 1–8.
• [19] M. Lin, Remarks on two recent results of Audenaert, Linear Algebra Appl. 489 (2016), 24–29.
• [20] L. Zou and Y. Jiang, A note on interpolation between the arithmetic-geometric mean and Cauchy-Schwarz matrix norm inequalities, J. Math. Inequal. 10 (2016), no. 4, 1119–1122.