On inequalities of Bellman and Aczél type

1.   Introduction

Let $\mathbb{B}\left(\mathcal{H} \right)$ be the ${{C}^{*}}$- algebra of all bounded linear operators on a Hilbert space $\mathcal{H}$. When $\mathcal{H}$ is finite dimensional of dimension $n$, we identify $\mathbb{B}\left(\mathcal{H} \right)$ with ${{\mathcal{M}}_{n}}$; the algebra of all complex $n\times n$ matrices. Let ${{\mathscr{H}}_{n}}$ be the set of all Hermitian matrices in ${{\mathcal{M}}_{n}}$. We denote by ${{\mathscr{H}}_{n}}\left(J \right)$ the set of all Hermitian matrices in ${{\mathcal{M}}_{n}}$ whose spectra are contained in an interval $J\subseteq \mathbb{R}$. The set of all positive definite matrices in ${{\mathcal{M}}_{n}}$ is denoted by ${{\mathscr{P}}_{n}}$. By ${{I}_{n}}$ we denote the identity matrix of ${{\mathcal{M}}_{n}}$. We write $A\ge 0$ if $A$ is a positive semidefinite matrix, and $A > 0$ if $A\ge 0$ is invertible (or strictly positive definite). A linear map $\Phi :{{\mathcal{M}}_{n}}\to {{\mathcal{M}}_{m}}$ is said to be positive if $0\le \Phi \left(A \right)$ when $0\le A$. If, in addition, $\Phi \left({{I}_{n}} \right) = {{I}_{m}}$, it is said to be unital.

When $0\le t\le 1$, the arithmetic mean and geometric mean of $A, B > 0$ are defined and denoted by

Let $x = \left({{x}_{1}}, \ldots, {{x}_{n}} \right)$ be an element of ${{\mathbb{R}}^{n}}$. Let ${{x}^{\downarrow }}$ and ${{x}^{\uparrow }}$ be the vectors obtained by rearranging the coordinates of $x$ in decreasing and increasing order respectively. Thus $x_{1}^{\downarrow }\ge \cdots \ge x_{n}^{\downarrow }$ and $x_{1}^{\uparrow }\le \cdots \le x_{n}^{\uparrow }$. For $A\in {{\mathcal{M}}_{n}}$ with real eigenvalues, $\lambda \left(A \right)$ is a vector of the eigenvalues of $A$. Then, ${{\lambda }^{\downarrow }}\left(A \right)$ and ${{\lambda }^{\uparrow }}\left(A \right)$ can be defined as above. Let $x, y\in {{\mathbb{R}}^{n}}$. The weak majorization relation $x\; {{\prec }_{w}}\; y$ means $\sum\nolimits_{j = 1}^{k}{x_{j}^{\downarrow }}\le \sum\nolimits_{j = 1}^{k}{y_{j}^{\downarrow }}, \left(1\le k\le n \right)$. If further equality holds for $k = n$ then we have the majorization $x\; \prec \; y$. Similarly, the weak supermajorization relation $x\; {{\prec }^{w}}\; y$ means $\sum\nolimits_{j = 1}^{k}{x_{j}^{\uparrow }}\ge \sum\nolimits_{j = 1}^{k}{y_{j}^{\uparrow }}, \left(1\le k\le n \right)$.

Let $A\in {{\mathscr{H}}_{n}}\left(J \right)$ have spectral decomposition $A = {{U}^{*}}\text{diag}\left({{\lambda }_{1}}, {{\lambda }_{2}}, \ldots, {{\lambda }_{n}} \right)U,$ where $U$ is a unitary and ${{\lambda }_{1}}, {{\lambda }_{2}}, \ldots, {{\lambda }_{n}}$ are the eigenvalues of $A$. Let $f$ be a real valued function defined on $J$. Then $f\left(A \right)$ is defined by $f\left(A \right) = {{U}^{*}}\text{diag}\left(f\left({{\lambda }_{1}} \right), f\left({{\lambda }_{2}} \right), \ldots, f\left({{\lambda }_{n}} \right) \right)U$.

The scalar Bellman inequality ^[3] says that if $p$ is a positive integer and $a, b, {{a}_{i}}, {{b}_{i}}\left(1\le i\le n \right)$ are positive real numbers such that $\sum\nolimits_{i = 1}^{n}{a_{i}^{p}}\le {{a}^{p}}$ and $\sum\nolimits_{i = 1}^{n}{b_{i}^{p}}\le {{b}^{p}}$, then

A multiplicative analogue of this inequality is due to Aczél ^[1]. In 1956, he proved that

where ${{a}_{i}}, {{b}_{i}}\left(1\le i\le n \right)$ are positive real numbers such that $a_{1}^{2}-\sum\nolimits_{i = 2}^{n}{a_{i}^{2}} > 0$ or $b_{1}^{2}-\sum\nolimits_{i = 2}^{n}{b_{i}^{2}} > 0$.

The operator theory related to inequalities in Hilbert space is studied in many papers. In [11,Corollary 2.2], Morassaei et al. showed the following non-commutative version of classical Bellman inequality:

where $A, B\in \mathbb{B}\left(\mathcal{H} \right)$ are two contractions (i.e., $0\le A, B\le I$) and $\Phi :\mathbb{B}\left(\mathcal{H} \right)\to \mathbb{B}\left(\mathcal{H} \right)$ is a unital positive linear map. The reverse inequality holds when $\frac{1}{2}\le p\le 1$ or $p\le -1$ [13,Theorem 3]. Actually, this result follows from the following inequality

where $f$ is an operator convex.

In [12,Theorem 2.2], Moslehian noted the following inequality for non-negative operator decreasing and operator concave $f$ and $p, q > 1$ with $\frac{1}{p}+\frac{1}{q} = 1$:

This inequality may be considered as operator versions of Aczél inequality.

As it is mentioned in [6,Corollary 1.12], the function $f$ on $\left[ 0, \infty \right)$ is operator concave if and only if $f$ is operator monotone (increasing). So, the above inequality is valid just for the functions of type $f\left(t \right) = \alpha +\beta t$. In this paper, we give a matrix version of the inequality (1.1) for decreasing concave function $f$ on $\left[ 0, \infty \right)$. Let $f$ be a convex function (in the usual sense) on $J$, $A, B\in {{\mathscr{H}}_{n}}\left(J \right)$, and $0\le t\le 1$. In this paper we prove that the eigenvalues of $f\left(\Phi \left(A \right){{\nabla }_{t}}\Phi \left(B \right) \right)$ are weakly majorized by the eigenvalues of $\Phi \left(f\left(A \right) \right){{\nabla }_{t}}\Phi \left(f\left(B \right) \right)$. The results presented in this paper are motivated by the results in ^[2,10,12,13].

2.   Main results

We start from the well-known Jensen inequality [4,p. 281]: If $A\in {{\mathscr{H}}_{n}}\left(J \right)$ and $f$ is a convex (resp. concave) function on $J$, then for any $x\in {{\mathbb{C}}^{m}}$ with $\left\| x \right\| = 1$,

The following result provides an extension of (2.1).

Lemma 2.1. Let $\Phi :{{\mathcal{M}}_{n}}\to {{\mathcal{M}}_{m}}$ be a unital positive linear map, $f$ be a convex function on $J$, and $x\in {{\mathbb{C}}^{m}}$ with $\left\| x \right\| = 1$. Then

for all $A\in {{\mathscr{H}}_{n}}\left(J \right)$.

Proof. We know that if $f$ is a convex function on an interval $J$, then for each point $\left(s, f\left(s \right) \right)$, there exists a real number ${{C}_{s}}$ such that

(if $f$ is differentiable at $s$, then ${{C}_{s}} = f'\left(s \right)$.)

Fix $s\in J$. Since $J$ contains the spectra of the $A$, we may replace $t$ in the above inequality by $A$, via a functional calculus to get

Applying the positive linear mappings $\Phi$, this implies

The inequality (2.4) easily implies, for any $x\in {{\mathbb{C}}^{m}}$ with $\left\| x \right\| = 1$,

On the other hand, since $\Phi$ is unital, we have $\left\langle \Phi \left(A \right)x, x \right\rangle \in J$ where $x\in {{\mathbb{C}}^{m}}$ with $\left\| x \right\| = 1$. Therefore, we may replace $s$ by $\left\langle \Phi \left(A \right)x, x \right\rangle$ in (2.5). This yields (2.2).

Remark 2.2. From inequality (2.3) one can infer that

This inequality can be regarded as a reverse of (2.2).

Actually, inequality (2.2) implies

From the assumptions on $\Phi$, we can write

Consequently, for any unit vector $x\in {{\mathbb{C}}^{m}}$,

Now, (2.6) follows from (2.7) by putting $t = \left\langle \Phi \left(A \right)x, x \right\rangle$.

We repeat the following result from ^[7] for completeness.

Remark 2.3. Regarding the convexity of $f$ on the interval $J$, we have

for any $s, t\in J$ and $r = \min \left\{ v, 1-v \right\}$ with $0\le v\le \frac{1}{2}$. For the case $\frac{1}{2}\le v\le 1$, the same result is true.

Thus.

holds for any $s, t\in J$ and $r = \min \left\{ v, 1-v \right\}$ with $0\le v\le 1$.

From the above inequality one can write

Dividing by $v > 0$, we get

Now, if $v\to 0$, and by taking into account that for $0\le v\le \frac{1}{2}$, $r = v$ we infer

This result can be considered as a refinement of inequality (2.3). Thus, if we apply the same arguments as in Lemma 2.1, we can obtain a sharper estimate than (2.2). Namely,

Lemma 2.4. [4,p. 35] If ${\lambda }_{j}^{\downarrow }\left(A \right)$ denote the eigenvalues of $n\times n$ Hermitian matrix $A$ arranged in decreasing order, then

where the maximum is taken over all choices of orthonormal vectors ${{u}_{1}}, \ldots, {{u}_{k}}$.

Theorem 2.5. Let $\Phi :{{\mathcal{M}}_{n}}\to {{\mathcal{M}}_{m}}$ be a unital positive linear map, $f$ be a convex (resp. concave) function on $J$, and $0\le t\le 1$. Then

for all $A, B\in {{\mathscr{H}}_{n}}\left(J \right)$.

Proof. Since $x\; {{\prec }_{w}}\; y$ if and only if $\left(-x \right)\; {{\prec }^{w}}\; \left(-y \right)$, it suffices to consider the convex case. Let ${{\lambda }_{1}}, \ldots, {{\lambda }_{n}}$ be the eigenvalues of $\left(1-t \right)\Phi \left(A \right)+t\Phi \left(B \right)$ and let ${{u}_{1}}, \ldots, {{u}_{n}}$ be the corresponding orthonormal eigenvectors arranged such that $f\left({{\lambda }_{1}} \right)\ge \cdots \ge f\left({{\lambda }_{n}} \right)$. Let $k = 1, \ldots, n$. Then

Therefore, we conclude

so that we get the desired conclusion.

If we choose $\Phi \left(X \right) = X$ in Theorem 2.5, we recover [2,Theorem 2.3], i.e.,

Corollary 2.6. Let $\Phi :{{\mathcal{M}}_{n}}\to {{\mathcal{M}}_{m}}$ be a unital positive linear map, and $0\le t\le 1$, $p\ge 1$. Then

for all $A, B\in {{\mathscr{P}}_{n}}$ such that $0\le A, B\le {{I}_{n}}$. In particular, for all unitarily invariant norm ${{\left\| \cdot \right\|}_{u}}$,

Inequality (2.8) can be regarded as a weak majorization version of [13,Theorem 3].

Theorem 2.7. Let $f$ be a decreasing concave function on $\left[ 0, \infty \right)$ and $0\le t\le 1$. Then

for all $A, B\in {{\mathscr{P}}_{n}}$.

Proof. By the minimax principle, for any integer $j$ less than or equal to the dimension of the space, we have a subspace $\mathscr{F}$ of dimension $j$ such that

and hence we have (2.9).

Note that the above statement is equivalent to the existence of a unitary operator $U$ satisfying in the following inequality:

Inequality (2.10) yields inequality

Furthermore, if $AB = BA$ we can write

Corollary 2.8. Let $A, B\in {{\mathscr{P}}_{n}}$ be contractive (in the sense that $\left\| A \right\|, \left\| B \right\|\le 1$, where $\left\| \cdot \right\|$ is the usual operator norm). Then for $j = 1, \ldots, n$

Proof. This inequality follows immediately from Theorem 2.7 by choosing $f\left(x \right) = 1-x$ on $\left(0, 1 \right)$ and $t = {1}/{2}$.

Remark 2.9. It has been shown in ^[8] that if $A, B$ are contractive, then for $j = 1, \ldots, n$

We refer the reader to ^[9] for further results of this type of inequalities. If $A, B\in {{\mathscr{P}}_{n}}$ are contractive, this inequality implies that

We remark that there is no ordering between (2.11) and (2.12). To see this, letting $A = \left[ \begin{matrix} 0.4 & -0.1 \\ -0.1 & 0.9 \\ \end{matrix} \right]$ and $B = \left[ \begin{matrix} 0.2 & 0.2 \\ 0.2 & 0.7 \\ \end{matrix} \right]$. Direct computation shows that

and

Acknowledgement

The authors would like to express their hearty thanks to the referees for their valuable comments. This work was financially supported by Islamic Azad University, Mashhad Branch.

Conflict of interest

The authors declare that there is no interest regarding the publication of this paper.