Bounds on Spreads of Matrices related to Fourth Central Moment. II

TL;DR

This paper establishes new inequalities involving the first four central moments of distributions, providing bounds for eigenvalues, matrix spreads, and polynomial roots, with a focus on real eigenvalues.

Contribution

It introduces novel bounds on eigenvalues, matrix spreads, and polynomial roots based on fourth central moments, extending previous inequalities.

Findings

Bounds for eigenvalues and matrix spread derived.

Inequalities for roots and span of polynomials established.

Applicable to matrices with real eigenvalues.

Abstract

We derive some inequalities involving first four central moments of discrete and continuous distributions. Bounds for the eigenvalues and spread of a matrix are obtained when all its eigenvalues are real. Likewise, we discuss bounds for the roots and span of a polynomial equation.

Equations

$$\begin{array}[]{lcl}\left|\mu_{3}\right|\leq\frac{\left(M-m\right)^{3}}{6\sqrt{3}}\ \ \text{and}\ \ \mu_{4}\leq\frac{\left(M-m\right)^{4}}{12}.&&\end{array}$$

$$\begin{array}[]{lcl}\mu_{2}\leq\frac{\left(M-m\right)^{2}}{4}&&\end{array}$$

$$\begin{array}[]{lcl}\mu_{2}\leq\left(M-\mu_{1}^{\prime}\right)\left(\mu_{1}^{\prime}-m\right)&&\end{array}$$

$$\begin{array}[]{lcl}\frac{\mu_{2}^{2}-\left(\mu_{1}^{\prime}-m\right)^{2}\mu_{2}}{\mu_{1}^{\prime}-m}\leq\mu_{3}\leq\frac{\left(M-\mu_{1}^{\prime}\right)^{2}\mu_{2}-\mu_{2}^{2}}{M-\mu_{1}^{\prime}}&&\end{array}$$

$$\begin{array}[]{lcl}\mu_{2}+\left(\frac{\mu_{3}}{2\mu_{2}}\right)^{2}\leq\frac{\left(M-m\right)^{2}}{4}.&&\end{array}$$

$$\begin{array}[]{lcl}m_{r}=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\overline{x})^{r}&&\end{array}$$

$$\begin{array}[]{lcl}m_{2r}\geq\frac{1+\left(n-1\right)^{2r-1}}{n\left(n-1\right)^{2r-1}}\left(x_{j}-\overline{x}\right)^{2r}&&\end{array}$$

$$\begin{array}[]{lcl}m_{2}\leq\left(n-1\right)\left(M-\overline{x}\right)^{2}\text{ \ and \ }m_{2}\leq\left(n-1\right)\left(\overline{x}-m\right)^{2}.&&\end{array}$$

$$\begin{array}[]{lcl}\mu_{4}\leq\alpha\mu_{2}+\beta&&\end{array}$$

$$\begin{array}[]{lcl}\alpha=\frac{3}{4}\left(M-m\right)^{2}-2\left(\mu_{1}^{\prime}-m\right)\left(M-\mu_{1}^{\prime}\right)&&\end{array}$$

$$\begin{array}[]{lcl}\beta=\left(\mu_{1}^{\prime}-m\right)\left(M-\mu_{1}^{\prime}\right)\left(\frac{1}{4}\left(M-m\right)^{2}-\left(\mu_{1}^{\prime}-m\right)\left(M-\mu_{1}^{\prime}\right)\right).&&\end{array}$$

$${\left(y+\frac{a+b}{2}\right)}^2\left(y-a\right)\left(y-b\right)\le 0$$

$$\begin{array}[]{lcl}y_{i}^{4}\leq\left(\frac{3}{4}\left(a+b\right)^{2}-ab\right)y_{i}^{2}+\left(\frac{\left(a+b\right)^{3}}{4}-\left(a+b\right)ab\right)y_{i}-\frac{\left(a+b\right)^{2}}{4}ab&&\end{array}$$

$$\begin{array}[]{lcl}\sum\limits_{i=1}^{n}p_{i}y_{i}^{4}\leq\left(\frac{3}{4}\left(a+b\right)^{2}-ab\right)\sum\limits_{i=1}^{n}p_{i}y_{i}^{2}+\left(\frac{\left(a+b\right)^{3}}{4}-\left(a+b\right)ab\right)\sum\limits_{i=1}^{n}p_{i}y_{i}-\frac{\left(a+b\right)^{2}}{4}ab.&&\end{array}$$

$$\begin{array}[]{lcl}\mu_{4}^{\prime}\leq\left(\frac{3}{4}\left(m+M\right)^{2}-mM\right)\mu_{2}^{\prime}+\frac{1}{4}\left(m+M\right)\left(M-m\right)^{2}\mu_{1}^{\prime}-\frac{1}{4}mM\left(m+M\right)^{2}.&&\end{array}$$

$$\begin{array}[]{lcl}\mu_{4}+3\mu_{2}^{2}\leq\left(M-m\right)^{2}\left(\mu_{1}^{\prime}-m\right)\left(M-\mu_{1}^{\prime}\right)\leq\frac{\left(M-m\right)^{4}}{4}.&&\end{array}$$

$$\begin{array}[]{lcl}\mu_{4}+3\mu_{2}^{2}\leq\alpha\mu_{2}+\beta+3\mu_{2}^{2}.&&\end{array}$$

$$\begin{array}[]{lcl}\mu_{4}+3\mu_{2}^{2}\leq\alpha\left(M-\mu_{1}^{\prime}\right)\left(\mu_{1}^{\prime}-m\right)+\beta+3\left(M-\mu_{1}^{\prime}\right)^{2}\left(\mu_{1}^{\prime}-m\right)^{2}.&&\end{array}$$

$${\left(M-m\right)}^2\left({\mu }_1^{\prime }-m\right)\left(M-{\mu }_1^{\prime }\right)=p_1{p}_2{\left(M-m\right)}^4={\mu }_4+3{\mu }_2^2.$$

$${\mu }_2^2\le \frac{{\mu }_4+3{\mu }_2^2}{4}\le \frac{{\left(M-m\right)}^4}{16},$$

$$\begin{array}[]{lcl}m_{4}+3m_{2}^{2}\leq\frac{\left(M-m\right)^{4}}{4},&&\end{array}$$

$$\begin{array}[]{lcl}m_{4}+3m_{2}^{2}\leq\frac{n^{2}-1}{4n^{2}}\left(M-m\right)^{4}&&\end{array}$$

$$\begin{array}[]{lcl}m_{4}+3m_{2}^{2}\leq\left(M-m\right)^{2}\left(M-\overline{x}\right)\left(\overline{x}-m\right).&&\end{array}$$

$$\overline{x}=\frac{\left(n-1\right)m+\left(n+1\right)M}{2n}\text{ or }\overline{x}=\frac{\left(n+1\right)m+\left(n-1\right)M}{2n},$$

$$\begin{array}[]{lcl}\left(M-\overline{x}\right)\left(\overline{x}-m\right)\leq\frac{n^{2}-1}{4n^{2}}\left(M-m\right)^{2}.&&\end{array}$$

$$\begin{array}[]{lcl}m_{4}\leq\max_{j\in I}\frac{j\left(n-j\right)\left(n^{2}-3nj+3j^{2}\right)}{n^{4}}\left(M-m\right)^{4}.&&\end{array}$$

$${\left(x-\overline{x}\right)}^4\le \frac{{\left(M-\overline{x}\right)}^4-{\left(m-\overline{x}\right)}^4}{M-m}\left(x-\overline{x}\right)-\frac{\left(m-\overline{x}\right){\left(M-\overline{x}\right)}^4-{\left(m-\overline{x}\right)}^4\left(M-\overline{x}\right)}{M-m},$$

$$\begin{array}[]{lcl}m_{4}\leq\left(\overline{x}-m\right)\left(M-\overline{x}\right)\left(\left(\overline{x}-m\right)^{2}+\left(M-\overline{x}\right)^{2}-\left(\overline{x}-m\right)\left(M-\overline{x}\right)\right),&&\end{array}$$

$$\overline{x}=\frac{jm+\left(n-j\right)M}{n}\text{.}$$

$$\begin{array}[]{lcl}\left(\overline{x}-m\right)\left(M-\overline{x}\right)\left(\left(\overline{x}-m\right)^{2}+\left(M-\overline{x}\right)^{2}-\left(\overline{x}-m\right)\left(M-\overline{x}\right)\right)&&\\ =\frac{j\left(n-j\right)\left(n^{2}-3nj+3j^{2}\right)}{n^{4}}\left(M-m\right)^{4}.&&\end{array}$$

Full text

† AMS classification 60E15, 15A42, 12D10

1. Introduction

This is the continuation of the work of Sharma et al. [24]. It is shown in [24] that for both discrete and continuous random variable in $[m,M]$, we have

[TABLE]

The inequalities (1) and the related Popoviciu inequality [15],

[TABLE]

provide lower bounds for the range $r=M-m$ of the random variable in terms of its central moments.  These inequalities are also useful in many other contexts. In literature, such inequalities are used to derive lower bounds for the spread of a matrix, and span of a polynomial equation. The idea of the spread is due to Mirsky [11] and the notion of the span was introduced by Robinson [16]. For more detail and further related topic see [1, 5, 6, 8, 9, 10, 11, 12, 18, 19, 20, 21, 22, 23].

In the present context we also need the following inequalities, see [3, 13, 18, 19],

[TABLE]

[TABLE]

and

[TABLE]

A generalization of Samuelson’s inequality [17] due to Sharma and Saini [21] says that if $\overline{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}$ is the arithmetic mean and

[TABLE]

is the $r$-th central moment of $n$ real numbers $x_{1},x_{2},...,x_{n}$, then

[TABLE]

for all $j=1,2,...,n$ and $r=1,2,...$. For $r=1$, the inequality (7) corresponds to Samuelson’s inequality [17], and the complementary inequalities due to Brunk [7] assert that

[TABLE]

We here discuss some further extensions and applications of the above inequalities to the field of theory of polynomial equations and matrix analysis. Our first result gives an upper bound for $\mu_{4}$ in terms of $m,M,\mu_{1}^{\prime}\ \text{and}\ \mu_{2}$ (Theorem 2.1, below) and this provides further refinements of the inequality (2) (Corollary 2.2-2.3). An extension of the second inequality (1) is obtained for the fourth central moment $m_{4}$ (Theorem 2.4). One more lower bound for the range $r$ is obtained in terms of the second and fourth central moment (Theorem 2.5). Some refinements of the first inequality (1) are given (Theorem 2.6). It is show that the inequalities (4) yield some more inequalities involving second and third central moments (Theorem 2.7). The inequalities analogous to the inequality (8) involving second and fourth central moments are proved (Theorem 2.8, Corollary 2.9-2.10). The upper (lower) bound for the smallest (largest) eigenvalue of a complex $n\times n$ matrix are given when all its eigenvalues are real as in case of Hermitian matrix (Theorem 3.1). The lower bounds for the spread are obtained in terms of traces of $A^{r}$, $r=1,2,3,4$ (Theorem 3.3). Some results are extended for positive unital linear functionals (Theorem 3.5). Likewise, we discuss bounds for the roots and span of a polynomial equation (Theorem 4.1-4.2).

2. Main Results

It is enough to prove the following results for the case when $X$ is a discrete random variable taking finitely many values $x_{1},x_{2},...,x_{n}$ with probabilities $p_{1},p_{2},...,p_{n}\,$, respectively. The arguments are similar for the case when $X$ is a continuous random variable.

Theorem 2.1**.**

For $m\leq x_{i}\leq M,$ $i=1,2,...,n$, we have

[TABLE]

where

[TABLE]

and

[TABLE]

Proof.

Note that

[TABLE]

if and only if $a\leq y\leq b$. Applying this to $n$ numbers $y_{i}$’s, $a\leq y_{i}\leq b$; a little computation shows that

[TABLE]

for all $i=1,2,...,n$. Multiplying both sides of (12) by $p_{i}\geq 0$, and adding the $n$ resulting inequalities, we get that

[TABLE]

For y_{i}=x_{i}-\mu_{1}^{\prime},\we have $\sum\limits_{i=1}^{n}p_{i}y_{i}=0,$ $\sum\limits_{i=1}^{n}p_{i}y_{i}^{2}=\mu_{2}$, $\sum\limits_{i=1}^{n}p_{i}y_{i}^{4}=\mu_{4}$, a=m-\mu_{1}^{\prime}\ and $b=M-\mu_{1}^{\prime}$. Using these facts and choosing $y_{i}=x_{i}-\mu_{1}^{\prime}$ in (13); the inequality (9) follows immediately. ∎

Likewise, we can derive the upper bound for the fourth moment about origin in terms of first and second moment, we have

[TABLE]

Further, the inequality (14) and hence the inequality (9) becomes equality for $n=2$. In this case $\mu_{1}^{\prime}=mp_{1}+Mp_{2}$, $\mu_{2}^{\prime}=m^{2}p_{1}+M^{2}p_{2}$ and the right hand side expression (14) equals $\mu_{4}^{\prime}=m^{4}p_{1}+M^{4}p_{2}$.

Corollary 2.2**.**

With the conditions as in Theorem 2.1, we have

[TABLE]

Proof.

The inequality (9) implies that

[TABLE]

From (3) and (16), we get that

[TABLE]

Inserting the values of $\alpha$ and $\beta$ respectively from (10) and (11) in (17), and simplifying the resulting expression, we immediately get the first inequality (15). The second inequality (15) follows on using arithmetic mean-geometric mean inequality. ∎

For $n=2$, the first inequality (15) becomes equality. We have

[TABLE]

By (15), we also get the following refinement of the inequality (2) for both discrete and continuos distributions,

[TABLE]

cf. [22].

Let $m_{2}$ and $m_{4}$ respectively denote the second and fourth central moment of $x_{1},x_{2},...,x_{n}$ as defined in (6). Then, by (15),

[TABLE]

and equality holds for $n=2$. The inequality (18) may be strengthened when $n$ is odd.

Corollary 2.3**.**

For $m\leq x_{i}\leq M;i=1,2,...,n$, and with $n$ odd, the inequality

[TABLE]

holds true.

Proof.

By the first inequality (15),

[TABLE]

Let $f\left(x\right)=\left(x-m\right)\left(M-x\right)$. Then the function $f$ increases in $\left[m,\frac{m+M}{2}\right]$, decreases in $\left[\frac{m+M}{2},M\right]$ and attains its maximum at $x=\frac{m+M}{2}$. Thus, if $n$ is even the maximum value of $\left(\overline{x}-m\right)\left(M-\overline{x}\right)$ is $\frac{\left(M-m\right)^{2}}{4}$ and this occurs when $x_{1}=x_{2}=...=x_{\frac{n}{2}}=m$ and $x_{\frac{n}{2}+1}=x_{\frac{n}{2}+2}=...=x_{n}=M$. But when $n$ is odd $\overline{x}\neq$ $\frac{m+M}{2}$ and $\left(\overline{x}-m\right)\left(M-\overline{x}\right)$ achieves its maximum when $x_{1}=x_{2}=...=x_{\frac{n-1}{2}}=m$ and $x_{\frac{n+1}{2}}=...=x_{n}=M$ or $x_{1}=x_{2}=...=x_{\frac{n+1}{2}}=m$ and $x_{\frac{n+3}{2}}=...=x_{n}=M$. Thus, the maximum is achieved either at

[TABLE]

and in each case

[TABLE]

The inequality (19) now follows from (20) and (21). ∎

In a similar vein we now prove an extension of the second inequality (1) for the central moment $m_{4}$.

Theorem 2.4**.**

For $m\leq x_{i}\leq M,\ i=1,2,...,n$, and $I=\left\{1,2,...,n-1\right\}$, we have

[TABLE]

Proof.

Let $f\left(x\right)=\left(x-\overline{x}\right)^{4}$. The function $f$ is convex on the interval $\left[m-\overline{x},\ M-\overline{x}\right]$. Therefore, for $m\leq x\leq M$,

[TABLE]

with equality if and only if $x=m$ or $x=M$. On applying this to $n$ numbers $x_{i}$’s, $m\leq x_{i}\leq M$ and using the arguments as in the proof of Theorem 2.1, we obtain after simplification

[TABLE]

with equality if and only if $x_{1}=x_{2}=...=x_{j}=m$ and $x_{j+1}=...=x_{n}=M$, $j=1,2,...,n-1.$ Hence, for some $j\in I$, the right hand side (23) achieves its maximum at

[TABLE]

For this value of $\overline{x}$ we have

[TABLE]

The inequality (22) now follows from (23) and (24). ∎

Theorem 2.5**.**

For $m\leq x_{i}\leq M,\ i=1,2,...,n$, we have

[TABLE]

Proof.

We find from the inequality (9) that

[TABLE]

Combining (3) and (26), and inserting the values of $\alpha$ and $\beta$ respectively from (10) and (11), we get that

[TABLE]

The right hand side expression (27) achieves its maximum in the interval $m\leq$ $\mu_{1}^{\prime}\leq M$ at $\mu_{1}^{\prime}=\frac{2m+M}{3}$ and $\mu_{1}^{\prime}=\frac{2M+m}{3},$ where its value is $\frac{4}{3^{5}}\left(M-m\right)^{6}$. This proves the theorem. ∎

The inequality (25) becomes equality for $n=2$ with $p_{1}=\frac{1}{3}$ and $p_{2}=\frac{2}{3}$ or $p_{1}=\frac{2}{3}$ and $p_{2}=\frac{1}{3}$. In this case, $\mu_{1}^{\prime}=\frac{2m+M}{3}$ or $\mu_{1}^{\prime}=\frac{2M+m}{3}$, $\mu_{2}=\frac{2}{9}(M-m)^{2}$ and $\mu_{4}=\frac{2}{27}(M-m)^{4}$.

It may be noted here that for $n=3$, $m_{4}=\frac{3}{2}m_{2}^{2}$ and the inequalities (19), (22) and (25) become same, $m_{4}\leq\frac{2}{27}\left(M-m\right)^{4}$.

We need Pearson’s inequality [14] in the proof of the following theorem. This inequality gives a relation between skewness and kurtosis of a distribution and can be written in the form

[TABLE]

Theorem 2.6**.**

For $m\leq x_{i}\leq M,\ i=1,2,...,n$,  the inequalities

[TABLE]

and

[TABLE]

hold true.

Proof.

The first inequality (29) follows immediately from the inequality(28).

Let $f\left(x\right)=ax-x^{3}.$ Then $f\left(x\right)$ has maxima at $x=\sqrt{\frac{a}{3}}$ where its value is $\frac{2}{3\sqrt{3}}a^{\frac{3}{2}}.$ We thus have

[TABLE]

The inequality (31) together with the second inequality (1) yields the second inequality (29).

The first inequality (30) follows immediately from the inequality (5). The second inequality (30) follows from the fact that $f\left(x\right)=\left(M-m\right)^{2}x^{4}-4x^{6}$ has maximum at $x=\frac{M-m}{\sqrt{6}}$ and its maximum value is $\frac{\left(M-m\right)^{6}}{108}$. ∎

For $n=2$, $\mu_{3}^{2}=\mu_{2}\mu_{4}-\mu_{2}^{3}=p_{1}^{2}p_{2}^{2}(p_{1}-p_{2})^{2}(M-m)^{6}$ if and only if $p_{1}=\frac{1}{2}\pm\frac{1}{2\sqrt{3}}$ and $p_{2}=\frac{1}{2}\mp\frac{1}{2\sqrt{3}}$, and in this case inequalities (29) become equalities. Likewise, the inequalities (30) reduce to equalities for $n=2$.

From (31) and the first inequality (29), we find that

[TABLE]

This gives the following inequality involving skewness $\left(\alpha_{3}\right)$ and kurtosis $\left(\alpha_{4}\right)$,

[TABLE]

Further, from the first inequality (30), we have

[TABLE]

where $q$ is studentized range, $q=\frac{r}{\sqrt{\mu_{2}}}$.

Theorem 2.7**.**

For $m\leq x_{i}\leq M,\ i=1,2,...,n$, we have

[TABLE]

[TABLE]

[TABLE]

and for $0<m\leq x\leq M$,

[TABLE]

Proof.

From the first inequality (4), we get that

[TABLE]

Let $f\left(x\right)=x^{3}-k^{2}x^{2}$. Then $f\left(x\right)$ has minimum at $x=\frac{2}{3}k^{2}$ where its value is $-\frac{4}{27}k^{6}.$ Thus we have

[TABLE]

Combining (36) and (37); we immediately get the first inequality (32). The second inequality (32) follows similarly from the second inequality (4).

To prove (33) we write (4) in the form

[TABLE]

The inequalities (33)  follow from (38) and the fact that the right hand side and left hand side expressions (38) achieve their maxima and minima at $\mu_{2}^{\prime}=\frac{M\left(M+\mu_{1}^{\prime}\right)}{2}$ and $\mu_{2}^{\prime}=\frac{m\left(m+\mu_{1}^{\prime}\right)}{2}$, respectively.

Likewise, the inequalities (34) follow from (4) and the fact that the right hand side and left hand side expressions (4) achieve their maxima and minima at $\mu_{2}=\frac{\left(M-\mu_{1}^{\prime}\right)^{2}}{2}$ and $\mu_{2}=\frac{\left(\mu_{1}^{\prime}-m\right)^{2}}{2}$, respectively.

Further, from the first inequality (4) we have

[TABLE]

For $m\geq 0,$ the inequalities (35) and (39) are equivalent. ∎

The inequalities (8) give the upper bounds for $m_{2}$ in terms of the values of m,\ M\and $\overline{x}$. We prove analogous inequalities involving $m_{2}$ and $m_{4}$.

Theorem 2.8**.**

For $m\leq x_{i}\leq M,\ i=1,2,...,n$, the inequalities

[TABLE]

and

[TABLE]

hold true.

Proof.

For $r=2$, it follows from (7) that

[TABLE]

From (3), we have

[TABLE]

Combining (42) and (43) we immediately get (40). The inequality (41) follows on using similar arguments. ∎

The inequalities (8) can equivalently be written respectively in the form

[TABLE]

and are important in finding the upper (lower) bound for the smallest (largest) value of the data. We mention here analogous inequalities involving $m_{2}$ and $m_{4}.$

Corollary 2.9**.**

Under the conditions of the above theorem, we have

[TABLE]

and

[TABLE]

Proof.

The inequalities (45) and (46) follow immediately from the inequalities (40) and (41), respectively. ∎

The inequalities (45) and (46) give better estimates than the corresponding estimates given by the inequalities (44). Note that the inequality

[TABLE]

holds true if and only if

[TABLE]

This is true, see [24].

The well known Karl Pearson coefficient of dispersion $V=\frac{\sqrt{m_{2}}}{\overline{x}}$ is a widely used measure of dispersion. We mention a lower for the ratio $\frac{\alpha_{4}}{V^{4}}$.

Corollary 2.10**.**

For $x_{i}>0,\ i=1,2,...,n$, we have

[TABLE]

Proof.

If all the $x_{i}$’s are positive, $m\geq 0$, and (47) follows from (45). ∎

3. Bounds for eigenvalues and spreads of matrices

Let $\mathbb{M}(n)$ denote the algebra of all $n\times n$ complex matrices. The eigenvalues $\lambda\left(A\right)$ of an element $A\in\mathbb{M}(n)$ are the roots of the characteristic polynomial $\det\left(A-\lambda\left(A\right)I\right)=0$ and are difficult to evaluate in general. The bounds for eigenvalues have been studied extensively in literature, for example see [2, 18, 21, 25]. In particular, Wolkowicz and Styan [25] have shown that

[TABLE]

where $B=A-\frac{\text{tr}A}{n}I$.  We prove an extension of this result in the following theorem.

Theorem 3.1**.**

If the eigenvalues of $A\in\mathbb{M}(n)$ are all real, then

[TABLE]

and

[TABLE]

Proof.

The arithmetic mean of the eigenvalues $\lambda_{i}(A)$ can be written as $\overline{\lambda}\left(A\right)=\frac{1}{n}\sum_{i=1}^{n}\lambda_{i}(A)=\frac{\text{tr}A}{n}$. The second and fourth central moment of the eigenvalues can be expressed in terms of tr$B^{2}$ and tr$B^{4}$, respectively. We have

[TABLE]

Apply Corollary 2.9., the inequalities (49) and (50) follow on substituting the values of $m_{2}$ and $m_{4}$ from (51) and $\overline{x}=\overline{\lambda}\left(A\right)=\frac{\text{tr}A}{n}$ in (45) and (46), respectively. ∎

The ratio of the largest and smallest eigenvalue of a positive definite matrix $A$ is known as the ratio spread or condition number ($c\left(A\right)$) of a positive definite matrix. Wolkowicz and Styan [25] have shown that

[TABLE]

It follows from Theorem $3.1$ that if $A$ is positive definite, then

[TABLE]

Example 3.2**.**

Let

[TABLE]

From (48), $\lambda_{\min}(A_{1})\leq 3.8417$ and $\lambda\max(A_{1})\geq 7.1583$ while from (49) and (50) we have better estimates $\lambda_{\min}(A_{1})\leq 3.7414$ and $\lambda\max(A_{1})\geq 7.2586$, respectively. Also, from (52) and (53), we respectively have $c\left(A_{1}\right)\geq 1.8633$ and $c\left(A_{1}\right)\geq 1.9401$.

Theorem 3.3**.**

Under the conditions of Theorem $3.1$, we have

[TABLE]

When $n$ is odd the inequality (54) may be strengthened to

[TABLE]

Proof.

The inequalities (54) and (55) follow respectively from the inequalities (15) and (19) on using arguments similar to those used in the proof of Theorem 3.1. ∎

Likewise, from the inequality (22), we have

[TABLE]

where $I=\left\{1,2,...,n-1\right\}$.

Also, from (25), we have

[TABLE]

It is shown in [24] that

[TABLE]

For $n=3$, (55) and (56) are identical. The inequality (56) provides better estimate than (58). The inequalities (54) and (55) are clearly independent. We show by means of examples that (54), (57) and (58) are independent.

Example 3.4**.**

Let

[TABLE]

The eigenvalues of  $A_{2}$ are $1,\ 4,\ \frac{5}{2}\pm\frac{\sqrt{33}}{2}$. From (54) and (58), spd$(A_{1})\geq 6.0264$, spd$(A_{2})\geq 4.6734$ and spd$(A_{1})\geq 6.2303$, spd$(A_{2})\geq 4.5767$, respectively. So, the inequalities (54) and (58) are independent. From (57), spd$(A_{1})\geq 6.3068$ and spd$(A_{2})\geq 4.762$. But for the matrix $A_{3}$ with eigenvalues $-1,\ 0,1$ with respective multiplicities $2,\ 5$ and $2$, we have (57) and (58) from spd$(A_{3})\geq 1.5131$ and spd$(A_{3})\geq 1.5197$, respectively. From the inequality (56), we have spd$(A_{1})\geq 6.2549,$ spd$(A_{2})\geq 4.5948$ and spd$(A_{3})\geq 1.5902$. The inequality (56) makes an improvement on (58).

We now show that some of the above inequalities can be extended for positive linear functionals. A linear functional $\varphi:\mathbb{M}(n)\rightarrow\mathbb{C}$ is called positive if $\varphi\left(A\right)\geq 0$ whenever $A\geq O$ and unital if $\varphi\left(I\right)=1$, see [4, 24].

Theorem 3.5**.**

Let $\varphi:\mathbb{M}(n)\rightarrow\mathbb{C}$ be a positive unital linear functional and let $A$ be any Hermitian element of $\mathbb{M}(n)$. Then

[TABLE]

where $B=A-\varphi\left(A\right)I$.

Proof.

By the spectral theorem, for $r=1,2,...,$ we have

[TABLE]

where $P_{i}$ are corresponding projections, $P_{i}\geq O$ and $\sum_{i=1}^{n}P_{i}=I$.

On applying $\varphi$, we find from (60) that

[TABLE]

with $\sum_{i=1}^{n}\varphi\left(P_{i}\right)=1$.

Note that $\varphi\left(A\right),$ $\varphi\left(B^{2}\right)$ and $\varphi\left(B^{4}\right)$ are respectively the arithmetic mean, second and fourth central moments of the eigenvalues $\lambda_{i}\left(A\right)$ with respective weights $\varphi\left(P_{i}\right),\ i=1,2,...,n$. So, we can apply Corollary 2.2, and the inequality (59) follows from (15). ∎

On using similar arguments one can easily obtain from Theorem 2.5 that

[TABLE]

Further, it is shown in [24] that

[TABLE]

and by (29) we have the following refinement of this inequality

[TABLE]

4. Bounds for the roots and span of a polynomial

Some bounds for the roots and span of a polynomial with all its roots real are considered in [24]. We here obtain some more bounds for the roots and span in terms of the first five coefficients of the polynomial.

Let $x_{1},\ x_{2},...,x_{n}$ denote the roots of the monic polynomial equation

[TABLE]

We assume that all the roots of $f(x)$ are real. On using the relations between roots and coefficients of a polynomial, one can see that the arithmetic mean $\overline{x}$ of the $x_{i}$’s equals zero. The second and fourth central moment can respectively be written as, see [24],

[TABLE]

Theorem 4.1**.**

Let the roots of the polynomial (64) be all real. Denote by $x_{\min}(f)$ and $x_{\max}(f)$ the smallest and largest root of $f(x)=0$. Then, for $n\geq 5,$ we have

[TABLE]

and

[TABLE]

Proof.

Apply Corollary 2.9., the inequalities (66) and (67) follow on substituting the values of $m_{2}$ and $m_{4}$ from (65) and $\overline{x}=0$ in (45) and (46), respectively. ∎

Theorem 4.2**.**

If roots of the polynomial (64) are all real, then

[TABLE]

[TABLE]

and

[TABLE]

When $n$ is odd the inequality (68) may be strengthened to

[TABLE]

Proof.

On using arguments similar to those used in the proof of Theorem 4.1 the inequalities (68), (69), (70) and (71) follow respectively from (15), (22), (25) and (19). ∎

Acknowledgments The authors are grateful to Prof. Rajendra Bhatia for the useful discussions and suggestions, and first author thanks Ashoka University for a visit in January 2019. The support of UGC-SAP is acknowledged.