On Some Inequalities for the Generalized Euclidean Operator Radius

Abstract

In the literature, there are many criteria to generalize the concept of a numerical radius; one of the most recent and interesting generalizations is the so-called generalized Euclidean operator radius, which reads: ${\omega }_p\left(T_1,\cdots ,T_n\right):=\underset{∥x∥=1}{sup}{\left({\displaystyle \sum _{i=1}^n}{\left|⟨T_{i}x,x⟩\right|}^p\right)}^{1/p},p\ge 1,$ for all Hilbert space operators $T_1,\cdots ,T_n$. Simply put, it is the numerical radius of multivariable operators. This study establishes a number of new inequalities, extensions, and generalizations for this type of numerical radius. More precisely, by utilizing the mixed Schwarz inequality and the extension of Furuta’s inequality, some new refinement inequalities are obtained for the numerical radius of multivariable Hilbert space operators. In the case of $n=1$, the resulting inequalities could be considered extensions and generalizations of the classical numerical radius.

1. Introduction

Let $\mathcal{B}\left(\mathcal{H}\right)$ be the Banach algebra of all bounded linear operators defined on a complex Hilbert space $\left(\mathcal{H};⟨·,·⟩\right)$ with the identity operator $1_{\mathcal{H}}$ in $\mathcal{B}\left(\mathcal{H}\right)$.

The numerical range $W\left(T\right)$ of a bounded linear operator T on a Hilbert space $\mathcal{H}$ is the image of the unit sphere of $\mathcal{H}$ under the quadratic form $x\to ⟨Tx,x⟩$ associated with an operator. More precisely, we set

$$\begin{array}{c}\hfill W\left(T\right)=\left\{⟨Tx,x⟩:x\in \mathcal{H},∥x∥=1\right\}.\end{array}$$

Additionally, the numerical radius is defined by

$$\begin{array}{c}\hfill w\left(T\right)=sup\left\{\left|\lambda \right|:\lambda \in W\left(T\right)\right\}=\underset{∥x∥=1}{sup}\left|⟨Tx,x⟩\right|.\end{array}$$

We recall that the usual operator norm of an operator T is

$$\begin{array}{c}\hfill ∥T∥=sup\left\{∥Tx∥:x\in H,∥x∥=1\right\}.\end{array}$$

It is well known that $w\left(·\right)$ defines an operator norm on $\mathcal{B}\left(\mathcal{H}\right)$, which is equivalent to the operator norm $\parallel ·\parallel$. Moreover, the following inequalities hold:

$$\begin{array}{c}\hfill \frac{1}{2}\parallel T\parallel \le w\left(T\right)\le \parallel T\parallel \end{array}$$

(1)

for any $T\in \mathcal{B}\left(\mathcal{H}\right)$, and these inequalities are sharp.

It is known that $w\left(T\right)$ is a norm on $\mathcal{B}\left(\mathcal{H}\right)$, but it is not unitarily invariant. However, the numerical radius norm is weakly unitarily invariant, i.e., $w\left(U^{\ast }TU\right)=w\left(T\right)$ for all partial isometries U. Additionally, let us not miss the chance to mention the important property that $w\left(T\right)=w\left(T^{\ast }\right)$ and $w\left(T^{\ast }T\right)=w\left(T{T}^{\ast }\right)$ for every $T\in \mathcal{B}\left(\mathcal{H}\right)$.

Denote $\left|T\right|={\left(T^{\ast }T\right)}^{1/2}$ as the absolute value of the operator T. Then, we have

$$\begin{array}{c}\hfill w\left(\left|T\right|\right)=\parallel T\parallel .\end{array}$$

It is well known that the numerical radius is not submultiplicative, but the following inequality holds:

$$\begin{array}{c}\hfill w\left(TS\right)\le 4w\left(T\right)w\left(S\right)\end{array}$$

for all $T,S\in \mathcal{B}\left(\mathcal{H}\right)$. In particular, if T and S commute, then

$$\begin{array}{c}\hfill w\left(TS\right)\le 2w\left(T\right)w\left(S\right).\end{array}$$

Moreover, if T and S are normal, then $w\left(·\right)$ is submultiplicative, i.e.,

$$\begin{array}{c}\hfill w\left(TS\right)\le w\left(T\right)w\left(S\right).\end{array}$$

For other related inequalities regarding the numerical radius, the reader is recommended to refer to [1,2,3].

In 2009, Popescu [4] introduced the concept of the Euclidean operator radius of an n-tuple $\mathbf{T}=\left(T_1,\cdots ,T_n\right)\in \mathcal{B}{\left(\mathcal{H}\right)}^n:=\mathcal{B}\left(\mathcal{H}\right)\times \cdots \times \mathcal{B}\left(\mathcal{H}\right)$. Namely, for $T_1,\cdots ,T_n\in \mathcal{B}\left(\mathcal{H}\right)$, the Euclidean operator radius of $T_1,\cdots ,T_n$ is defined by

$$\begin{array}{c}\hfill w_{\mathrm{e}}\left(T_1,\cdots ,T_n\right):=\underset{∥x∥=1}{sup}{\left(\sum _{i=1}^n{\left|⟨T_{i}x,x⟩\right|}^2\right)}^{1/2}.\end{array}$$

Indeed, the Euclidean operator radius was generalized in [5] as follows:

$$\begin{array}{c}\hfill {\omega }_p\left(T_1,\cdots ,T_n\right):=\underset{∥x∥=1}{sup}{\left(\sum _{i=1}^n{\left|⟨T_{i}x,x⟩\right|}^p\right)}^{1/p},p\ge 1.\end{array}$$

If $p=1$, then $w_1\left(T_1,\cdots ,T_n\right)$ (also denoted by $w_R\left(T_1,\cdots ,T_n\right)$) is called the Rhombic numerical radius, which has been studied in [6]. The following case is of particular interest: $w_1\left(C,\cdots ,C\right)=n·w\left(C\right)$. In fact, the generalized operator radius is a natural generalization of the concept of a numerical radius. Since the progress in this area is rapidly increasing, we recommend the recent results concerning the numerical radius and norm inequalities [7,8,9,10,11,12,13,14,15,16,17,18,19] and the references therein, where other related results are discussed as well.

We note that in the case of $p=\infty$, the generalized Euclidean operator radius is defined as

$$\begin{array}{cc}\hfill w_{\infty }\left(T_1,\cdots ,T_n\right)& :=\underset{p\to \infty }{lim}{\omega }_p\left(T_1,\dots ,T_n\right)=\underset{∥x∥=1}{sup}\underset{1\le i\le n}{max}\left|⟨T_{i}x,x⟩\right|.\hfill \end{array}$$

Thus, the following inequalities hold:

$$\begin{array}{c}\hfill w_{\infty }\left(T_1,\cdots ,T_n\right)\le {\omega }_p\left(T_1,\cdots ,T_n\right)\le w_R\left(T_1,\cdots ,T_n\right)\end{array}$$

(2)

for all $p\ge 1$. This fact follows with Jensen’s inequality applied for the function $h\left(p\right)={\omega }_p\left(T_1,\cdots ,T_n\right)$, which is log-convex and decreasing for all $p>1$.

On the other hand, let us recall the following Jensen inequality:

$$\begin{array}{c}\hfill {\left(\frac{1}{n}\sum _{k=1}^n{a}_k\right)}^p\le \frac{1}{n}\sum _{k=1}^n{a}_k^p,\end{array}$$

which holds for every finite sequence of positive real numbers ${\left(a_k\right)}_{k=1}^n$ and $p\ge 1$. Hence, by setting $a_k=\left|⟨T_{k}x,x⟩\right|$ for all $(k=1,\cdots ,n)$, we obtain

$$\begin{array}{c}\hfill \sum _{k=1}^n\left|⟨T_{k}x,x⟩\right|\le n^{1-\frac{1}{p}}{\left(\sum _{k=1}^n{\left|⟨T_{k}x,x⟩\right|}^p\right)}^{\frac{1}{p}}.\end{array}$$

Taking the supremum over all unit vectors $x\in \mathcal{H}$, we get

$$\begin{array}{c}\hfill w_R\left(T_1,\cdots ,T_n\right)\le n^{1-\frac{1}{p}}{\omega }_p\left(T_1,\cdots ,T_n\right).\end{array}$$

(3)

Combining the inequalities (2) and (3), we obtain

$$\begin{array}{c}\hfill w_{\infty }\left(T_1,\cdots ,T_n\right)\le {\omega }_p\left(T_1,\cdots ,T_n\right)\le w_R\left(T_1,\cdots ,T_n\right)\le n^{1-\frac{1}{p}}{\omega }_p\left(T_1,\cdots ,T_n\right).\end{array}$$

(4)

More generally, in the power mean inequality

$$\begin{array}{c}\hfill {\left(\frac{1}{n}\sum _{k=1}^n{a}_k^p\right)}^{\frac{1}{p}}\le {\left(\frac{1}{n}\sum _{k=1}^n{a}_k^q\right)}^{\frac{1}{q}},\end{array}$$

for all $p\le q$, if we choose $a_k=\left|⟨T_{k}x,x⟩\right|$ for all $(k=1,\cdots ,n)$, then we have

$$\begin{array}{c}\hfill {\left(\frac{1}{n}\sum _{k=1}^n{\left|⟨T_{k}x,x⟩\right|}^p\right)}^{\frac{1}{p}}\le {\left(\frac{1}{n}\sum _{k=1}^n{\left|⟨T_{k}x,x⟩\right|}^q\right)}^{\frac{1}{q}}.\end{array}$$

Taking the supremum over all unit vectors $x\in \mathcal{H}$, we obtain

$$\begin{array}{c}\hfill {\omega }_p\left(T_1,\cdots ,T_n\right)\le n^{\frac{1}{p}-\frac{1}{q}}{w}_q\left(T_1,\cdots ,T_n\right)\end{array}$$

(5)

for all $q\ge p\ge 1$. Indeed, one can refine (3) by applying the following Jensen inequality:

$$\begin{array}{c}\hfill {\left(\frac{1}{n}\sum _{k=1}^n{a}_k\right)}^p\le \frac{1}{n}\sum _{k=1}^n{a}_k^p-\frac{1}{n}\sum _{k=1}^n{\left|a_k-\frac{1}{n}\sum _{j=1}^n{a}_j\right|}^p,p\ge 2,\end{array}$$

(6)

which is obtained from a more general result for superquadratic functions [20]. For more results about operator inequalities involving superquadratic functions, see [1,15,16,17,21,22].

Thus, by setting $a_k=\left|⟨T_{k}x,x⟩\right|$ in (6), we obtain

$$\begin{array}{c}\hfill {\left(\sum _{k=1}^n\left|⟨T_{k}x,x⟩\right|\right)}^p\le n^{p-1}\sum _{k=1}^n{\left|⟨T_{k}x,x⟩\right|}^p-n^{p-1}\sum _{k=1}^n{\left|\left|⟨T_{k}x,x⟩\right|-\frac{1}{n}\sum _{j=1}^n\left|⟨T_{j}x,x⟩\right|\right|}^p.\end{array}$$

Taking the supremum again over all unit vectors $x\in \mathcal{H}$, we obtain

$$\begin{array}{cc}\hfill & \underset{∥x∥=1}{sup}{\left(\sum _{k=1}^n\left|⟨T_{k}x,x⟩\right|\right)}^p\hfill \\ & \le \underset{∥x∥=1}{sup}\left\{n^{p-1}\sum _{k=1}^n{\left|⟨T_{k}x,x⟩\right|}^p-n^{p-1}\sum _{k=1}^n{\left|\left|⟨T_{k}x,x⟩\right|-\frac{1}{n}\sum _{j=1}^n\left|⟨T_{j}x,x⟩\right|\right|}^p\right\}\hfill \\ \hfill & \le n^{p-1}\underset{∥x∥=1}{sup}\sum _{k=1}^n{\left|⟨T_{k}x,x⟩\right|}^p-n^{p-1}\underset{∥x∥=1}{inf}\sum _{k=1}^n{\left|\left|⟨T_{k}x,x⟩\right|-\frac{1}{n}\sum _{j=1}^n\left|⟨T_{j}x,x⟩\right|\right|}^p,\hfill \end{array}$$

which gives

$$\begin{array}{c}\hfill w_R^p\left(T_1,\cdots ,T_n\right)\le n^{p-1}{\omega }_p^p\left(T_1,\cdots ,T_n\right)-n^{p-1}\underset{∥x∥=1}{inf}\sum _{k=1}^n{\left|\left|⟨T_{k}x,x⟩\right|-\frac{1}{n}\sum _{j=1}^n\left|⟨T_{j}x,x⟩\right|\right|}^p.\end{array}$$

The obtained bound refines the right-hand side of (4). Clearly, all the above-mentioned inequalities generalize and refine some inequalities obtained in [23]. For recent inequalities, counterparts, refinements, and other related properties concerning the generalized Euclidean operator radius, the reader may refer to [4,5,6,23,24,25,26,27,28].

In this work, inequalities for the generalized operator radius are presented. Our proofs are based on the Dragomir extension of the Furuta inequality and the mixed Schwarz inequality (or the so-called Kato inequality) for Hilbert space operators in $\mathcal{B}\left(\mathcal{H}\right)$. It is proved that our obtained results extend and refine the inequalities given in [5,23,26,27]. In fact, the presented inequalities generalize and extend those given in [5,23], particularly.

2. Bounds for the Generalized Euclidean Operator Radius

Before stating our main results, some useful lemmas on standard inequalities are necessary to present.

Lemma 1.

The inequalities below hold [29].

1. 

The Power-Mean inequality:

$$\begin{array}{c}\hfill a^{\alpha }{b}^{1-\alpha }\le \alpha a+\left(1-\alpha \right)b\le {\left(\alpha a^p+\left(1-\alpha \right)b^p\right)}^{\frac{1}{p}}\end{array}$$

(7)

for all $\alpha \in \left[0,1\right]$, $a,b\ge 0$ and $p\ge 1$.

2. 

The Power-Young inequality:

$$\begin{array}{c}\hfill ab\le \frac{a^{\alpha }}{\alpha }+\frac{b^{\beta }}{\beta }\le {\left(\frac{a^{p\alpha }}{\alpha }+\frac{b^{p\beta }}{\beta }\right)}^{\frac{1}{p}}\end{array}$$

(8)

for all $a,b\ge 0$ and $\alpha ,\beta >1$ with $\frac{1}{\alpha }+\frac{1}{\beta }=1$ and all $p\ge 1$.

Lemma 2.

(The McCarthy inequality) [30] Let $A\in \mathcal{B}{\left(\mathcal{H}\right)}^{+}$, then

$$\begin{array}{c}\hfill {⟨Ax,x⟩}^p\le ⟨A^{p}x,x⟩,p\ge 1,\end{array}$$

(9)

for any unit vector $x\in \mathcal{H}$. This inequality was refined and improved in the recent work [1].

Lemma 3.

If $a,b>0$, and $p,q>1$ such that $\frac{1}{p}+\frac{1}{q}=1,$ then for ($m=1,2,\cdots$), the following inequality holds [31]:

$$\begin{array}{c}\hfill {(a^{\frac{1}{p}}{b}^{\frac{1}{q}})}^m+r_0^m{(a^{\frac{m}{2}}-b^{\frac{m}{2}})}^2\le {\left(\frac{a^r}{p}+\frac{b^r}{q}\right)}^{\frac{m}{r}},r\ge 1,\end{array}$$

(10)

where $r_0=min\left\{\frac{1}{p},\frac{1}{q}\right\}$. In particular, if $p=q=2$, then

$${(a^{\frac{1}{2}}{b}^{\frac{1}{2}})}^m+\frac{1}{2^m}{(a^{\frac{m}{2}}-b^{\frac{m}{2}})}^2\le 2^{-\frac{m}{r}}{\left(a^r+b^r\right)}^{\frac{m}{r}}.$$

For $m=1$, it is reduces to

$$(a^{\frac{1}{2}}{b}^{\frac{1}{2}})+\frac{1}{2}{(a^{\frac{1}{2}}-b^{\frac{1}{2}})}^2\le 2^{-\frac{1}{r}}{\left(a^r+b^r\right)}^{\frac{1}{r}}.$$

In 1994, Furuta [32] proved the following generalization of Kato’s inequality (3):

$$\begin{array}{c}\hfill {\left|⟨T{\left|T\right|}^{\alpha +\beta -1}x,y⟩\right|}^2\le ⟨{\left|T\right|}^{2\alpha }x,x⟩⟨{\left|T\right|}^{2\beta }y,y⟩\end{array}$$

(11)

for any $x,y\in \mathcal{H}$ and $\alpha ,\beta \in \left[0,1\right]$ with $\alpha +\beta \ge 1$.

Dragomir in [33] generalized the inequality (11) for any $\alpha ,\beta \ge 0$ with $\alpha +\beta \ge 1$. Indeed, as noted by Dragomir, the condition $\alpha ,\beta \in \left[0,1\right]$ was assumed by Furuta to fit with the Heinz–Kato inequality, which reads as follows:

$$\begin{array}{c}\hfill \left|⟨Tx,y⟩\right|\le ∥A^{\alpha }x∥∥B^{1-\alpha }y∥\end{array}$$

for any $x,y\in \mathcal{H}$ and $\alpha \in \left[0,1\right]$ where A and B are positive operators, such that $∥Tx∥\le ∥Ax∥$ and $∥T^{\ast }y∥\le ∥By∥$ for any $x,y\in \mathcal{H}$.

In the same work [33], Dragomir provides a useful extension of Furuta’s inequality, described as follows:

$$\begin{array}{c}\hfill {\left|⟨DCBAx,y⟩\right|}^2\le ⟨A^{\ast }{\left|B\right|}^{2}Ax,x⟩⟨D{\left|C^{\ast }\right|}^2{D}^{\ast }y,y⟩\end{array}$$

(12)

for any $A,B,C,D\in \mathcal{B}\left(\mathcal{H}\right)$ and any vectors $x,y\in \mathcal{H}$. The equality in (12) holds if the vectors $BAx$ and $C^{\ast }{D}^{\ast }y$ are linearly dependent in $\mathcal{H}$. For other closely related versions of Kato’s inequality and its refinements, see [1,3,10,28,34,35,36,37,38].

2.1. Basic Properties of the Generalized Euclidean Operator Radius

Moslehian et al. [23] mention without proofs the following properties of the generalized Euclidean operator radius:

1.

${\omega }_p\left(T_1,\cdots ,T_n\right)=0$ if and only if $T_k=0$ for each ($k=1,\cdots ,n$);

2.

${\omega }_p\left(\lambda T_1,\cdots ,\lambda T_n\right)=\left|\lambda \right|{\omega }_p\left(T_1,\cdots ,T_n\right)$;

3.

${\omega }_p\left(A_1+B_1,\cdots ,A_n+B_n\right)\le {\omega }_p\left(A_1,\cdots ,A_n\right)+{\omega }_p\left(B_1,\cdots ,B_n\right)$;

4.

${\omega }_p\left(X^{\ast }{T}_{1}X,\cdots ,X^{\ast }{T}_{n}X\right)\le ∥X∥{\left({\displaystyle \sum _{i=1}^n}{\omega }^p\left(T_i\right)\right)}^{1/p}$

for every $T_k,A_k,B_k,X\in \mathcal{B}\left(\mathcal{H}\right)$$(k=1,\cdots ,n)$ and every scalar $\lambda \in \mathbb{C}$.

Despite the fact that the authors in [23] mentioned the above basic properties of the generalized Euclidean operator radius, it seems they missed some other important properties rather than leaving these properties without proof. Sometimes, it is nice to elucidate the proof of these elementary facts. As a result, we are going to give proof of each property. Clearly, the first two properties follow from the definition of the generalized Euclidean operator radius. In what follows, some classical properties are presented.

Let $T_1,\cdots ,T_n,U\in \mathcal{B}\left(\mathcal{H}\right)$ such that U is a a partial isometry. Then, the following properties of the generalized Euclidean operator radius hold:

1.

The generalized Euclidean operator radius is weakly unitarily invariant, i.e.,

$$\begin{array}{c}\hfill {\omega }_p\left(U^{\ast }{T}_{1}U,\cdots ,U^{\ast }{T}_{n}U\right)={\omega }_p\left(T_1,\cdots ,T_n\right);\end{array}$$

2.

${\omega }_p\left(T_1,\cdots ,T_n\right)={\omega }_p\left(T_1^{\ast },\cdots ,T_n^{\ast }\right)$;

3.

${\omega }_p\left(T_1^{\ast }{T}_1,\cdots ,T_n^{\ast }{T}_n\right)={\omega }_p\left(T_1{T}_1^{\ast },\cdots ,T_n{T}_n^{\ast }\right)$.

Proof. 

1.

The first property follows since

$$\begin{array}{cc}\hfill {\omega }_p\left(U^{\ast }{T}_{1}U,\cdots ,U^{\ast }{T}_{n}U\right)& :=\underset{∥x∥=1}{sup}{\left(\sum _{i=1}^n{\left|⟨U^{\ast }{T}_{i}Ux,x⟩\right|}^p\right)}^{1/p}\hfill \\ \hfill & =\underset{∥x∥=1}{sup}{\left(\sum _{i=1}^n{\left|⟨T_{i}Ux,Ux⟩\right|}^p\right)}^{1/p}\hfill \\ \hfill & =\underset{∥y∥=1}{sup}{\left(\sum _{i=1}^n{\left|⟨T_{i}y,y⟩\right|}^p\right)}^{1/p}\hfill \\ \hfill & ={\omega }_p\left(T_1,\cdots ,T_n\right).\hfill \end{array}$$

2.

By the definition of the generalized Euclidean operator radius, we have

$$\begin{array}{cc}\hfill {\omega }_p\left(T_1,\cdots ,T_n\right):=\underset{∥x∥=1}{sup}{\left(\sum _{i=1}^n{\left|⟨T_{i}x,x⟩\right|}^p\right)}^{1/p}& =\underset{∥x∥=1}{sup}{\left(\sum _{i=1}^n{\left|⟨x,T_i^{\ast }x⟩\right|}^p\right)}^{1/p}\hfill \\ \hfill & ={\omega }_p\left(T_1^{\ast },\cdots ,T_n^{\ast }\right).\hfill \end{array}$$

3.

Similarly, by definition and since $T_k{T}_k^{\ast }$ are selfadjoint operators for all ($k=1,\cdots ,n$), then we have

$$\begin{array}{cc}\hfill {\omega }_p\left(T_1{T}_1^{\ast },\cdots ,T_n{T}_n^{\ast }\right):=\underset{∥x∥=1}{sup}{\left(\sum _{i=1}^n{\left|⟨T_i{T}_i^{\ast }x,x⟩\right|}^p\right)}^{1/p}& =\underset{∥x∥=1}{sup}{\left(\sum _{i=1}^n{\left|⟨x,T_i{T}_i^{\ast }x⟩\right|}^p\right)}^{1/p}\hfill \\ \hfill & ={\omega }_p\left(T_1^{\ast }{T}_1,\cdots ,T_n^{\ast }{T}_n\right).\hfill \end{array}$$

4.

Finally, employing the classical Minkowski inequality, we obtain

$$\begin{array}{cc}\hfill {\omega }_p\left(A_1+B_1,\cdots ,A_n+B_n\right)& ={\left(\sum _{i=1}^n{\left|⟨\left(A_i+B_i\right)x,x⟩\right|}^p\right)}^{\frac{1}{p}}\hfill \\ \hfill & \le {\left(\sum _{i=1}^n{\left(\left|⟨A_{i}x,x⟩\right|+\left|⟨B_{i}x,x⟩\right|\right)}^p\right)}^{\frac{1}{p}}\hfill \\ \hfill & \le {\left(\sum _{i=1}^n{\left|⟨A_{i}x,x⟩\right|}^p\right)}^{\frac{1}{p}}+{\left(\sum _{i=1}^n{\left|⟨B_{i}x,x⟩\right|}^p\right)}^{\frac{1}{p}}\hfill \\ \hfill & \le {\omega }_p\left(A_1,\cdots ,A_n\right)+{\omega }_p\left(B_1,\cdots ,B_n\right),\hfill \end{array}$$

It remains to prove that ${\omega }_p\left(X^{\ast }{T}_{1}X,\cdots ,X^{\ast }{T}_{n}X\right)\le ∥X∥{({\displaystyle \sum _{i=1}^n}{\omega }^p\left(T_i\right))}^{1/p}$. From the definition of the generalized Euclidean operator radius, we have

$$\begin{array}{cc}\hfill {\omega }_p\left(X^{\ast }{T}_{1}X,\cdots ,X^{\ast }{T}_{n}X\right)& :=\underset{∥x∥=1}{sup}{\left(\sum _{i=1}^n{\left|⟨X^{\ast }{T}_{i}Xx,x⟩\right|}^p\right)}^{1/p}\hfill \\ \hfill & =\underset{∥x∥=1}{sup}{\left(\sum _{i=1}^n{\left|⟨T_{i}Xx,Xx⟩\right|}^p\right)}^{1/p}\hfill \\ \hfill & \le \underset{∥x∥=1}{sup}{\left(\sum _{i=1}^n{∥Xx∥}^p{\omega }^p\left(T_i\right)\right)}^{1/p}(\mathrm{By} \mathrm{Def}. \mathrm{of} \omega \left(·\right))\hfill \\ \hfill & \le \underset{∥x∥=1}{sup}{\left(\sum _{i=1}^n{∥X∥}^p{\omega }^p\left(T_i\right)\right)}^{1/p}\hfill \\ \hfill & =∥X∥{\left(\sum _{i=1}^n{\omega }^p\left(T_i\right)\right)}^{1/p}\hfill \end{array}$$

as required, which proves the last property. □

Proposition 1.

Let $T_1,\cdots ,T_n\in \mathcal{B}\left(\mathcal{H}\right)$ and $f,g$ be nonnegative continuous functions defined on $\left[0,\infty \right)$, satisfying that $f\left(t\right)g\left(t\right)=t$$(t\ge 0)$. Then, we have

$$\begin{array}{c}\hfill w_r^r\left(T_1,\cdots ,T_n\right)\le {\omega }_p\left(f^r\left(\left|T_1\right|\right),\cdots ,f^r\left(\left|T_n\right|\right)\right)w_q\left(g^r\left(\left|T_1^{\ast }\right|\right),\cdots ,g^r\left(\left|T_n^{\ast }\right|\right)\right)\end{array}$$

(13)

for all $r\ge 2$ and $p,q>1$ with $\frac{1}{p}+\frac{1}{q}=1$.

Proof. 

By the generalized Cauchy–Schwarz inequality [36], we have

$$\begin{array}{cc}\hfill {\left|⟨T_{i}x,x⟩\right|}^r& \le {∥f\left(\left|T_i\right|\right)x∥}^r{∥g\left(\left|T_i^{\ast }\right|\right)x∥}^r\hfill \\ \hfill & ={⟨f^2\left(\left|T_i\right|\right)x,x⟩}^{\frac{r}{2}}{⟨g^2\left(\left|T_i^{\ast }\right|\right)x,x⟩}^{\frac{r}{2}}\hfill \\ \hfill & \le ⟨f^r\left(\left|T_i\right|\right)x,x⟩⟨g^r\left(\left|T_i^{\ast }\right|\right)x,x⟩,(\mathrm{by} \mathrm{the} \mathrm{McCarthy} \mathrm{inequality}).\hfill \end{array}$$

Taking the sum over all i from 1 to n, we obtain

$$\begin{array}{cc}\hfill \sum _{i=1}^n{\left|⟨T_{i}x,x⟩\right|}^r& \le \sum _{i=1}^n⟨f^r\left(\left|T_i\right|\right)x,x⟩⟨g^r\left(\left|T_i^{\ast }\right|\right)x,x⟩\hfill \\ \hfill & \le {\left(\sum _{i=1}^n{⟨f^r\left(\left|T_i\right|\right)x,x⟩}^p\right)}^{\frac{1}{p}}{\left(\sum _{i=1}^n{⟨g^r\left(\left|T_i^{\ast }\right|\right)x,x⟩}^q\right)}^{\frac{1}{q}},(\mathrm{by} \mathrm{the} \mathrm{H}\text{ö}\mathrm{lder} \mathrm{inequality}).\hfill \end{array}$$

Taking the supremum over all vectors $x\in \mathcal{H}$ such that $\parallel x\parallel =1$, we obtain the desired result. □

Some interesting non-trivial special cases ramified from (13) are considered in the following corollaries.

Corollary 1.

Let $T_1,\cdots ,T_n\in \mathcal{B}\left(\mathcal{H}\right)$ and $f,g$ be nonnegative continuous functions defined on $\left[0,\infty \right)$, satisfying that $f\left(t\right)g\left(t\right)=t$$(t\ge 0)$. Then, we have

$$\begin{array}{c}\hfill w_r^r\left(T_1,\cdots ,T_n\right)\le w_{\mathrm{e}}\left(f^r\left(\left|T_1\right|\right),\cdots ,f^r\left(\left|T_n\right|\right)\right)w_{\mathrm{e}}\left(g^r\left(\left|T_1^{\ast }\right|\right),\cdots ,g^r\left(\left|T_n^{\ast }\right|\right)\right)\end{array}$$

(14)

for all $r\ge 2$.

Proof. 

It is enough to substitute $p=q=2$ in (13). □

Applying Furuta’s inequality, we can establish the following result.

Proposition 2.

Let $A_i,B_i,C_i,D_i\in \mathcal{B}\left(\mathcal{H}\right)$$(i=1,\cdots ,n)$. Then, we have

$$\begin{array}{c}{w}_{\mathrm{e}}^2\left(D_1{C}_1{B}_1{A}_1,\cdots ,D_n{C}_n{B}_n{A}_n\right)\hfill \\ \le {\omega }_p\left(A_1^{\ast }{\left|B_1\right|}^2{A}_1,\cdots ,A_n^{\ast }{\left|B_n\right|}^2{A}_n\right)w_q\left(D_1{\left|C_1^{\ast }\right|}^2{D}_1^{\ast },\cdots ,D_n{\left|C_n^{\ast }\right|}^2{D}_n^{\ast }\right)\hfill \end{array}$$

(15)

for all $p,q>1$ with $\frac{1}{p}+\frac{1}{q}=1$.

Proof. 

By setting $y=x$ in (12), we obtain

$$\begin{array}{cc}\hfill & \underset{∥x∥=1}{sup}\sum _{i=1}^n{\left|⟨D_i{C}_i{B}_i{A}_{i}x,x⟩\right|}^2\hfill \\ \hfill & \le \underset{∥x∥=1}{sup}\sum _{i=1}^n⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,x⟩⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,x⟩\hfill \\ \hfill & \le \underset{∥x∥=1}{sup}{\left(\sum _{i=1}^n{⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,x⟩}^p\right)}^{\frac{1}{p}}\underset{∥x∥=1}{sup}{\left(\sum _{i=1}^n{⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,x⟩}^q\right)}^{\frac{1}{q}}(\mathrm{by} \mathrm{the} \mathrm{H}\text{ö}\mathrm{lder} \mathrm{inequality})\hfill \\ \hfill & ={\omega }_p\left(A_1^{\ast }{\left|B_1\right|}^2{A}_1,\cdots ,A_n^{\ast }{\left|B_n\right|}^2{A}_n\right)w_q\left(D_1{\left|C_1^{\ast }\right|}^2{D}_1^{\ast },\cdots ,D_n{\left|C_n^{\ast }\right|}^2{D}_n^{\ast }\right),\hfill \end{array}$$

which gives the desired result. □

Corollary 2.

Let $B_i\in \mathcal{B}\left(\mathcal{H}\right)$$(i=1,\cdots ,n)$. Then, we have

$$\begin{array}{c}\hfill w_{\mathrm{e}}^2\left(B_1^2,\cdots ,B_n^2\right)\le {\omega }_p\left({\left|B_1\right|}^2,\cdots ,{\left|B_n\right|}^2\right)w_q\left({\left|B_1\right|}^2,\cdots ,{\left|B_n\right|}^2\right)\end{array}$$

(16)

for all $p,q>1$ such that $\frac{1}{p}+\frac{1}{q}=1$. In particular, for $p=q=2$, we have

$$\begin{array}{c}\hfill w_{\mathrm{e}}\left(B_1^2,\cdots ,B_n^2\right)\le w_{\mathrm{e}}\left({\left|B_1\right|}^2,\cdots ,{\left|B_n\right|}^2\right).\end{array}$$

(17)

Proof. 

Let us set $A_i=U_i$, $D_i=U_i^{\ast }$ ($U_i$ is a partial isometry) for all ($i=1,\cdots ,n$) and $C_i=B_i$ in the previous result. Then, we have

$$\begin{array}{c}\hfill w_{\mathrm{e}}^2\left(U_1^{\ast }{B}_1^2{U}_1,\cdots ,U_2^{\ast }{B}_n^2{U}_2\right)\le {\omega }_p\left(U_1^{\ast }{\left|B_1\right|}^2{U}_1,\cdots ,U_n^{\ast }{\left|B_n\right|}^2{U}_n\right)w_q\left(U_1^{\ast }{\left|B_1^{\ast }\right|}^2{U}_1,\cdots ,U_n^{\ast }{\left|B_n^{\ast }\right|}^2{U}_n\right)\end{array}$$

for all $p,q>1$ with $\frac{1}{p}+\frac{1}{q}=1$. Since ${\omega }_p\left(·\right)$ is weakly unitarily invariant and

$$\begin{array}{c}\hfill w_q\left({\left|B_1^{\ast }\right|}^2,\cdots ,{\left|B_n^{\ast }\right|}^2\right)=w_q\left({\left|B_1\right|}^2,\cdots ,{\left|B_n\right|}^2\right),\end{array}$$

the desired result is obtained. □

An interesting application of Furuta’s inequality is applied in the following result.

Corollary 3.

Let $T_i\in \mathcal{B}\left(\mathcal{H}\right)$$(i=1,\cdots ,n)$, and $\alpha ,\beta \ge 0$ such that $\alpha +\beta \ge 1$. Then, we have

$$\begin{array}{c}\hfill w_{\mathrm{e}}^2\left(T_1{\left|T_1\right|}^{\alpha +\beta -1},\cdots ,T_n{\left|T_n\right|}^{\alpha +\beta -1}\right)\le {\omega }_p\left({\left|T_1\right|}^{2\alpha },\cdots ,{\left|T_n\right|}^{2\alpha }\right)w_q\left({\left|T_1^{\ast }\right|}^{2\beta },\cdots ,{\left|T_n^{\ast }\right|}^{2\beta }\right)\end{array}$$

(18)

for all $p,q>1$ such that $\frac{1}{p}+\frac{1}{q}=1$.

Proof. 

Let $T_i=U_i\left|T_i\right|$ be the polar decomposition for ($i=1,\cdots ,n$). Let us set $D_i=U_i$, $B=1_{\mathcal{H}}$, $C={\left|T_i\right|}^{\beta }$ and $A_i={\left|T_i\right|}^{\alpha }$ for all $\alpha ,\beta \ge 0$ such that $\alpha +\beta \ge 1$ in (15). Then, we have

$$\begin{array}{c}\hfill D_i{C}_i{B}_i{A}_i=U_i{\left|T_i\right|}^{\beta }{\left|T_i\right|}^{\alpha }=U_i\left|T_i\right|{\left|T_i\right|}^{\alpha +\beta -1}=T_i{\left|T_i\right|}^{\alpha +\beta -1}.\end{array}$$

In addition, we have $A_i^{\ast }{\left|B_i\right|}^2{A}_i={\left|T_i\right|}^{2\alpha }$ and $D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }=U{\left|T_i\right|}^{2\beta }{U}^{\ast }={\left|T_i^{\ast }\right|}^{2\beta }$ for all ($i=1,\cdots ,n$). □

Corollary 4.

Let $T_i\in \mathcal{B}\left(\mathcal{H}\right)$$(i=1,\cdots ,n)$. Then, we have

$$\begin{array}{c}\hfill w_{\mathrm{e}}^2\left(T_1,\cdots ,T_n\right)\le {\omega }_p\left(\left|T_1\right|,\cdots ,\left|T_n\right|\right)w_q\left(\left|T_1^{\ast }\right|,\cdots ,\left|T_n^{\ast }\right|\right)\end{array}$$

(19)

for all $p,q>1$ such that $\frac{1}{p}+\frac{1}{q}=1$.

Proof. 

It is enough to put $\alpha =\beta =\frac{1}{2}$ in (18). □

Corollary 5.

Let $T_i\in \mathcal{B}\left(\mathcal{H}\right)$$(i=1,\cdots ,n)$, and $\alpha ,\beta \ge 0$ such that $\alpha +\beta \ge 1$. Then, we have

$$\begin{array}{c}\hfill w_{\mathrm{e}}^2\left(T_1\left|T_1\right|,\cdots ,T_n\left|T_n\right|\right)\le {\omega }_p\left({\left|T_1\right|}^2,\cdots ,{\left|T_n\right|}^2\right)w_q\left({\left|T_1\right|}^2,\cdots ,{\left|T_n\right|}^2\right)\end{array}$$

(20)

for all $p,q>1$ such that $\frac{1}{p}+\frac{1}{q}=1$. In particular, for $p=q=2$, we have

$$\begin{array}{c}\hfill w_{\mathrm{e}}^2\left(T_1\left|T_1\right|,\cdots ,T_n\left|T_n\right|\right)\le w_{\mathrm{e}}^2\left({\left|T_1\right|}^2,\cdots ,{\left|T_n\right|}^2\right).\end{array}$$

(21)

Proof. 

The proof follows by setting $\alpha =\beta =1$ in (18) and uses the properties of ${\omega }_p\left(·\right)$. □

2.2. Inequalities for the Generalized Euclidean Operator Radius

In this section, we provide some extended inequalities given in [5,23]. More precisely, all presented results in [23] are very special cases of the established inequalities.

Theorem 1.

Let $A_i,B_i,C_i,D_i\in \mathcal{B}\left(\mathcal{H}\right)$$(i=1,\cdots ,n)$. Then, we have

$$\begin{array}{cc}\hfill w_R\left(D_1{C}_1{B}_1{A}_1,\cdots ,D_n{C}_n{B}_n{A}_n\right)& \le ∥\sum _{i=1}^n\frac{1}{2p}\left[{\left(A_i^{\ast }{\left|B_i\right|}^2{A}_i\right)}^{\left(1-\gamma \right)p}+{\left(A_i^{\ast }{\left|B_i\right|}^2{A}_i\right)}^{\gamma p}\right]\hfill \\ \hfill & +\frac{1}{2q}\left[{\left(D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right)}^{\gamma q}+{\left(D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right)}^{\left(1-\gamma \right)q}\right]∥\hfill \end{array}$$

(22)

for all $p,q>1$ and $\gamma \in \left[0,1\right]$ such that $p-1\ge p\gamma \ge 1$, $q-1\ge q\gamma \ge 1$ and $\frac{1}{p}+\frac{1}{q}=1$.

Proof. 

Let us consider $y=x$ in (12). Then, we have

$$\begin{array}{cc}\hfill & \sum _{i=1}^n\left|⟨D_i{C}_i{B}_i{A}_{i}x,x⟩\right|\hfill \\ \hfill & \le \sum _{i=1}^n{⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,x⟩}^{\frac{1}{2}}{⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,x⟩}^{\frac{1}{2}}\hfill \\ \hfill & \le \frac{1}{2}\sum _{i=1}^n\left[{⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,x⟩}^{1-\gamma }{⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,x⟩}^{\gamma }\right.\left(\sin \mathrm{ce}\sqrt{ab}\le \frac{a^{\gamma }{b}^{1-\gamma }+a^{1-\gamma }{b}^{\gamma }}{2}\right)\hfill \\ \hfill & \left.+{⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,x⟩}^{\gamma }{⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,x⟩}^{1-\gamma }\right]\hfill \\ \hfill & \le \frac{1}{2}\sum _{i=1}^n\left\{{\left({⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,x⟩}^{\left(1-\gamma \right)p}+{⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,x⟩}^{\gamma p}\right)}^{\frac{1}{p}}\right.(\mathrm{by} \mathrm{the} \mathrm{H}\text{ö}\mathrm{lder} \mathrm{inequality})\hfill \\ \hfill & \left.\times {\left({⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,x⟩}^{\gamma q}+{⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,x⟩}^{\left(1-\gamma \right)q}\right)}^{\frac{1}{q}}\right\}\hfill \\ \hfill & \le \frac{1}{2p}\sum _{i=1}^n\left(⟨{\left(A_i^{\ast }{\left|B_i\right|}^2{A}_i\right)}^{\left(1-\gamma \right)p}x,x⟩+⟨{\left(A_i^{\ast }{\left|B_i\right|}^2{A}_i\right)}^{\gamma p}x,x⟩\right)\hfill \\ \hfill & +\frac{1}{2q}\sum _{i=1}^n\left(⟨{\left(D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right)}^{\gamma q}x,x⟩+⟨{\left(D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right)}^{\left(1-\gamma \right)q}x,x⟩\right)\hfill \\ \hfill & =\sum _{i=1}^n⟨\left\{\frac{1}{2p}\left[{\left(A_i^{\ast }{\left|B_i\right|}^2{A}_i\right)}^{\left(1-\gamma \right)p}+{\left(A_i^{\ast }{\left|B_i\right|}^2{A}_i\right)}^{\gamma p}\right]\right.\hfill \\ \hfill & \left.+\frac{1}{2q}\left[{\left(D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right)}^{\gamma q}+{\left(D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right)}^{\left(1-\gamma \right)q}\right]\right\}x,x⟩,\hfill \end{array}$$

where we have used the AM-GM and the McCarthy inequalities in the last inequality. Taking the supremum over all unit vectors $x\in \mathcal{H}$, we obtain the required result. □

Corollary 6.

Let $A_i,B_i,C_i,D_i\in \mathcal{B}\left(\mathcal{H}\right)$$(i=1,\cdots ,n)$. Then, we have

$$\begin{array}{c}\hfill w_R\left(D_1{C}_1{B}_1{A}_1,\cdots ,D_n{C}_n{B}_n{A}_n\right)\le \frac{1}{2}∥\sum _{i=1}^n\left[A_i^{\ast }{\left|B_i\right|}^2{A}_i+D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right]∥.\end{array}$$

Proof. 

It is enough to substitute $p=q=2$ in (22), implying that $\gamma =\frac{1}{2}$. □

Corollary 7.

Let $T_i\in \mathcal{B}\left(\mathcal{H}\right)$$(i=1,\cdots ,n)$. Then, we have

$$\begin{array}{c}{w}_R\left(T_1{\left|T_1\right|}^{\alpha +\beta -1},\cdots ,T_n{\left|T_n\right|}^{\alpha +\beta -1}\right)\hfill \\ \hfill \le ∥\sum _{i=1}^n\left\{\frac{1}{2p}\left[{\left|T_i\right|}^{2\alpha \left(1-\gamma \right)p}+{\left|T_i\right|}^{2\alpha \gamma p}\right]+\frac{1}{2q}\left[{\left|T_i^{\ast }\right|}^{2\beta \gamma q}+{\left|T_i^{\ast }\right|}^{2\beta \left(1-\gamma \right)q}\right]\right\}∥\end{array}$$

(23)

for all $p,q>1$ and $\gamma \in \left[0,1\right]$ such that $p-1\ge p\gamma \ge 1$, $q-1\ge q\gamma \ge 1$ and $\frac{1}{p}+\frac{1}{q}=1$.

In particular, we have

$$\begin{array}{c}\hfill w_R\left(T_1{\left|T_1\right|}^{\alpha +\beta -1},\cdots ,T_n{\left|T_n\right|}^{\alpha +\beta -1}\right)\le \frac{1}{2}∥\sum _{i=1}^n\left[{\left|T_i\right|}^{2\alpha }+{\left|T_i^{\ast }\right|}^{2\beta }\right]∥.\end{array}$$

(24)

Proof. 

The proof is similar to the one of the inequality (18) by employing (22). □

Theorem 2.

Let $A_i,B_i,C_i,D_i\in \mathcal{B}\left(\mathcal{H}\right)$$(i=1,\cdots ,n)$. Then, we have

$$\begin{array}{c}{\omega }_p^p\left(D_1{C}_1{B}_1{A}_1,\cdots ,D_n{C}_n{B}_n{A}_n\right)\hfill \\ \hfill \le \frac{1}{2^p}∥\sum _{i=1}^n{\left[\left(1-\gamma \right)A_i^{\ast }{\left|B_i\right|}^2{A}_i+\gamma D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right]}^p+{\left[\gamma A_i^{\ast }{\left|B_i\right|}^2{A}_i+\left(1-\gamma \right)D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right]}^p∥\end{array}$$

(25)

for all $\gamma \in \left[0,1\right]$ and $p\ge 1$. In particular, we have

$$\begin{array}{c}\hfill {\omega }_p^p\left(D_1{C}_1{B}_1{A}_1,\cdots ,D_n{C}_n{B}_n{A}_n\right)\le \frac{1}{2^{p-1}}∥\sum _{i=1}^n{\left(A_i^{\ast }{\left|B_i\right|}^2{A}_i+D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right)}^p∥.\end{array}$$

(26)

Proof. 

Let $y=x$ in (12). Then, we have

$$\begin{array}{cc}\hfill & \sum _{i=1}^n{\left|⟨D_i{C}_i{B}_i{A}_{i}x,x⟩\right|}^p\hfill \\ \hfill & \le \sum _{i=1}^n{\left({⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,x⟩}^{\frac{1}{2}}{⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,x⟩}^{\frac{1}{2}}\right)}^p\hfill \\ \hfill & \le \frac{1}{2^p}\sum _{i=1}^n\left[\left({⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,x⟩}^{\left(1-\gamma \right)}{⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,x⟩}^{\gamma }\right)\right.\left(\sin \mathrm{ce}\sqrt{ab}\le \frac{a^{\gamma }{b}^{1-\gamma }+a^{1-\gamma }{b}^{\gamma }}{2}\right)\hfill \\ \hfill & {\left.+\left({⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,x⟩}^{\gamma }{⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,x⟩}^{\left(1-\gamma \right)}\right)\right]}^p\hfill \\ \hfill & \le \frac{1}{2^p}\sum _{i=1}^n\left[{\left(\left(1-\gamma \right)⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,x⟩+\gamma ⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,x⟩\right)}^p\right.(\mathrm{by} \mathrm{the} \mathrm{AM}-\mathrm{GM} \mathrm{inequality})\hfill \\ \hfill & \left.+{\left(\gamma ⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,x⟩+\left(1-\gamma \right)⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,x⟩\right)}^p\right]\hfill \\ \hfill & =\frac{1}{2^p}\sum _{i=1}^n{⟨\left[\left(1-\gamma \right)A_i^{\ast }{\left|B_i\right|}^2{A}_i+\gamma D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right]x,x⟩}^p\hfill \\ \hfill & +\frac{1}{2}\sum _{i=1}^n{⟨\left[\gamma A_i^{\ast }{\left|B_i\right|}^2{A}_i+\left(1-\gamma \right)D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right]x,x⟩}^p\hfill \\ \hfill & \le \frac{1}{2^p}\sum _{i=1}^n⟨{\left[\left(1-\gamma \right)A_i^{\ast }{\left|B_i\right|}^2{A}_i+\gamma D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right]}^{p}x,x⟩(\mathrm{by} \mathrm{the} \mathrm{McCarthy} \mathrm{inequality})\hfill \\ \hfill & +\frac{1}{2}\sum _{i=1}^n⟨{\left[\gamma A_i^{\ast }{\left|B_i\right|}^2{A}_i+\left(1-\gamma \right)D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right]}^{p}x,x⟩\hfill \\ \hfill & =\frac{1}{2^p}⟨\sum _{i=1}^n\left\{{\left[\left(1-\gamma \right)A_i^{\ast }{\left|B_i\right|}^2{A}_i+\gamma D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right]}^p\right\}\hfill \\ \hfill & \left.+{\left[\gamma A_i^{\ast }{\left|B_i\right|}^2{A}_i+\left(1-\gamma \right)D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right]}^{p}x,x⟩\right].\hfill \end{array}$$

Taking the supremum over all unit vectors $x\in \mathcal{H}$, we obtain the required result. The particular case is obtained by setting $\gamma =\frac{1}{2}$ in (25). □

Corollary 8.

Let $T_i\in \mathcal{B}\left(\mathcal{H}\right)$$(i=1,\cdots ,n)$. Then, for all $\alpha ,\beta \ge 0$ such that $\alpha +\beta \ge 1$, we have

$$\begin{array}{c}\hfill {\omega }_p^p\left(T_1{\left|T_1\right|}^{\alpha +\beta -1},\cdots ,T_n{\left|T_n\right|}^{\alpha +\beta -1}\right)\le \frac{1}{2^{p-1}}∥\sum _{i=1}^n{\left({\left|T_i\right|}^{2\alpha }+{\left|T_i^{\ast }\right|}^{2\beta }\right)}^p∥\end{array}$$

(27)

for all $p\ge 1$.

Proof. 

Let $T_i=U_i\left|T_i\right|$ be the polar decomposition for ($i=1,\cdots ,n$). Setting $D_i=U_i$, $B=1_{\mathcal{H}}$, $C={\left|T_i\right|}^{\beta }$ and $A_i={\left|T_i\right|}^{\alpha }$ for all $\alpha ,\beta \ge 0$ such that $\alpha +\beta \ge 1$ in (26), then we obtain

$$\begin{array}{c}\hfill D_i{C}_i{B}_i{A}_i=U_i{\left|T_i\right|}^{\beta }{\left|T_i\right|}^{\alpha }=U_i\left|T_i\right|{\left|T_i\right|}^{\alpha +\beta -1}=T_i{\left|T_i\right|}^{\alpha +\beta -1}.\end{array}$$

In addition, we have $A_i^{\ast }{\left|B_i\right|}^2{A}_i={\left|T_i\right|}^{2\alpha }$ and $D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }=U_i{\left|T_i\right|}^{2\beta }{U}_i^{\ast }={\left|T_i^{\ast }\right|}^{2\beta }$ for all ($i=1,\cdots ,n$). □

The following result generalizes and extends the results proved in [5].

Theorem 3.

Let $A_i,B_i,C_i,D_i\in \mathcal{B}\left(\mathcal{H}\right)$$(i=1,\cdots ,n)$. Then, for $m\in \mathbb{N}$ and $r,p\ge m\ge 1$, we have

$$\begin{array}{c}{\omega }_p^p\left(D_1{C}_1{B}_1{A}_1,\cdots ,D_n{C}_n{B}_n{A}_n\right)\hfill \\ \hfill \le \frac{n^{1-\frac{m}{r}}}{2^{\frac{m}{r}}}{∥\sum _{i=1}^n\left({\left(A_i^{\ast }{\left|B_i\right|}^2{A}_i\right)}^{\frac{rp}{m}}+{\left(D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right)}^{\frac{rp}{m}}\right)∥}^{\frac{m}{r}}-\underset{∥x∥=1}{inf}{\xi }_{p,m}\left(x\right),\end{array}$$

(28)

where

$$\begin{array}{c}\hfill {\xi }_{p,m}\left(x\right):=2^{-m}\sum _{i=1}^n{\left({⟨{\left(A_i^{\ast }{\left|B_i\right|}^2{A}_i\right)}^{\frac{p}{m}}x,x⟩}^{\frac{m}{2}}-{⟨{\left(D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right)}^{\frac{p}{m}}x,x⟩}^{\frac{m}{2}}\right)}^2.\end{array}$$

Proof. 

Let $x\in \mathcal{H}$ be a unit vector. By setting $y=x$ in (12), we obtain

$$\begin{array}{cc}\hfill & \sum _{i=1}^n{\left|⟨D_i{C}_i{B}_i{A}_{i}x,x⟩\right|}^p\hfill \\ \hfill & \le \sum _{i=1}^n{⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,x⟩}^{\frac{p}{2}}{⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,x⟩}^{\frac{p}{2}}\hfill \\ \hfill & =\sum _{i=1}^n{\left({⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,x⟩}^{\frac{p}{2m}}{⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,x⟩}^{\frac{p}{2m}}\right)}^m\hfill \\ \hfill & \le \sum _{i=1}^n{\left({⟨{\left(A_i^{\ast }{\left|B_i\right|}^2{A}_i\right)}^{\frac{p}{m}}x,x⟩}^{\frac{1}{2}}{⟨{\left(D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right)}^{\frac{p}{m}}x,x⟩}^{\frac{1}{2}}\right)}^m(\mathrm{by} \mathrm{the} \mathrm{McCarthy} \mathrm{inequality})\hfill \\ \hfill & \le \sum _{i=1}^n{\left(\frac{1}{2}{⟨{\left(A_i^{\ast }{\left|B_i\right|}^2{A}_i\right)}^{\frac{p}{m}}x,x⟩}^r+\frac{1}{2}{⟨{\left(D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right)}^{\frac{p}{m}}x,x⟩}^r\right)}^{\frac{m}{r}}(\mathrm{by} (10\left)\right)\hfill \\ \hfill & -2^{-m}\sum _{i=1}^n{\left({⟨{\left(A_i^{\ast }{\left|B_i\right|}^2{A}_i\right)}^{\frac{p}{m}}x,x⟩}^{\frac{m}{2}}-{⟨{\left(D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right)}^{\frac{p}{m}}x,x⟩}^{\frac{m}{2}}\right)}^2\hfill \\ \hfill & \le \sum _{i=1}^n{\left(\frac{1}{2}⟨{\left(A_i^{\ast }{\left|B_i\right|}^2{A}_i\right)}^{\frac{rp}{m}}x,x⟩+\frac{1}{2}⟨{\left(D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right)}^{\frac{rp}{m}}x,x⟩\right)}^{\frac{m}{r}}(\mathrm{by} \mathrm{the} \mathrm{McCarthy} \mathrm{inequality})\hfill \\ \hfill & -2^{-m}\sum _{i=1}^n{\left({⟨{\left(A_i^{\ast }{\left|B_i\right|}^2{A}_i\right)}^{\frac{p}{m}}x,x⟩}^{\frac{m}{2}}-{⟨{\left(D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right)}^{\frac{p}{m}}x,x⟩}^{\frac{m}{2}}\right)}^2\hfill \\ \hfill & \le \frac{n^{1-\frac{m}{r}}}{2^{\frac{m}{r}}}\sum _{i=1}^n{\left(⟨{\left(A_i^{\ast }{\left|B_i\right|}^2{A}_i\right)}^{\frac{rp}{m}}x,x⟩+⟨{\left(D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right)}^{\frac{rp}{m}}x,x⟩\right)}^{\frac{m}{r}}(\mathrm{by} \mathrm{concavity} \mathrm{of} t^{\frac{m}{r}})\hfill \\ \hfill & -2^{-m}\sum _{i=1}^n{\left({⟨{\left(A_i^{\ast }{\left|B_i\right|}^2{A}_i\right)}^{\frac{p}{m}}x,x⟩}^{\frac{m}{2}}-{⟨{\left(D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right)}^{\frac{p}{m}}x,x⟩}^{\frac{m}{2}}\right)}^2.\hfill \end{array}$$

Taking the supremum over all unit vectors $x\in \mathcal{H}$, we obtain the required result. □

A very interesting result concerning n-tuples of operators $\left(T_1,\cdots ,T_n\right)$, which generalizes and extends the main result in [5], is given as follows:

Theorem 4.

Let $T_i\in \mathcal{B}\left(\mathcal{H}\right)$$(i=1,\cdots ,n)$, $\alpha ,\beta \ge 0$ such that $\alpha +\beta \ge 1$, $m\in \mathbb{N}$ and $r,p\ge m\ge 1$. Then, we have

$$\begin{array}{c}\hfill {\omega }_p^p\left(T_1{\left|T_1\right|}^{\alpha +\beta -1},\cdots ,T_n{\left|T_n\right|}^{\alpha +\beta -1}\right)\le \frac{n^{1-\frac{m}{r}}}{2^{\frac{m}{r}}}{∥\sum _{i=1}^n\left({\left|T_i\right|}^{\frac{2\alpha rp}{m}}+{\left|T_i^{\ast }\right|}^{\frac{2\beta rp}{m}}\right)∥}^{\frac{m}{r}}-\underset{∥x∥=1}{inf}{\psi }_{p,m,\alpha ,\beta }\left(x\right),\end{array}$$

(29)

where

$$\begin{array}{c}\hfill {\psi }_{p,m,\alpha ,\beta }\left(x\right):=2^{-m}\sum _{i=1}^n{\left({⟨{\left|T_i\right|}^{\frac{2\alpha p}{m}}x,x⟩}^{\frac{m}{2}}-{⟨{\left|T_i^{\ast }\right|}^{\frac{2\beta p}{m}}x,x⟩}^{\frac{m}{2}}\right)}^2.\end{array}$$

Proof. 

Let $T_i=U_i\left|T_i\right|$ be the polar decomposition for ($i=1,\cdots ,n$). Setting $D_i=U_i$, $B_i=1_{\mathcal{H}}$, $C_i={\left|T_i\right|}^{\beta }$ and $A_i={\left|T_i\right|}^{\alpha }$ for all ($i=1,\cdots ,n$) and all $\alpha ,\beta \ge 0$ such that $\alpha +\beta \ge 1$ in (28), then we have

$$\begin{array}{c}\hfill D_i{C}_i{B}_i{A}_i=U_i{\left|T_i\right|}^{\beta }{\left|T_i\right|}^{\alpha }=U_i\left|T_i\right|{\left|T_i\right|}^{\alpha +\beta -1}=T_i{\left|T_i\right|}^{\alpha +\beta -1}.\end{array}$$

Furthermore, we have $A_i^{\ast }{\left|B_i\right|}^2{A}_i={\left|T_i\right|}^{2\alpha }$ and $D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }=U_i{\left|T_i\right|}^{2\beta }{U}_i^{\ast }={\left|T_i^{\ast }\right|}^{2\beta }$ for all ($i=1,\cdots ,n$). □

Corollary 9.

Let $T_i\in \mathcal{B}\left(\mathcal{H}\right)$$(i=1,\cdots ,n)$, $\alpha ,\beta \ge 0$ such that $\alpha +\beta \ge 1$ and $p\ge 1$. Then, we have

$$\begin{array}{c}\hfill {\omega }_p^p\left(T_1{\left|T_1\right|}^{\alpha +\beta -1},\cdots ,T_n{\left|T_n\right|}^{\alpha +\beta -1}\right)\le \frac{1}{2}∥\sum _{i=1}^n\left({\left|T_i\right|}^{2\alpha p}+{\left|T_i^{\ast }\right|}^{2\beta p}\right)∥-\underset{∥x∥=1}{inf}{\psi }_{p,1,\alpha ,\beta }\left(x\right),\end{array}$$

(30)

where

$$\begin{array}{c}\hfill {\psi }_{p,1,\alpha ,\beta }\left(x\right):=\frac{1}{2}\sum _{i=1}^n{\left({⟨{\left|T_i\right|}^{2\alpha p}x,x⟩}^{\frac{1}{2}}-{⟨{\left|T_i^{\ast }\right|}^{2\beta p}x,x⟩}^{\frac{1}{2}}\right)}^2.\end{array}$$

Proof. 

The proof is obtained by setting $m=r=1$ in (29). □

Corollary 10.

Let $T_i\in \mathcal{B}\left(\mathcal{H}\right)$$(i=1,\cdots ,n)$. Then, for $m\in \mathbb{N}$ and $r,p\ge m\ge 1$,

$$\begin{array}{c}\hfill {\omega }_p^p\left(T_1,\cdots ,T_n\right)\le \frac{n^{1-\frac{m}{r}}}{2^{\frac{m}{r}}}{∥\sum _{i=1}^n\left({\left|T_i\right|}^{\frac{rp}{m}}+{\left|T_i^{\ast }\right|}^{\frac{rp}{m}}\right)∥}^{\frac{m}{r}}-\underset{∥x∥=1}{inf}{\psi }_{p,m,\frac{1}{2},\frac{1}{2}}\left(x\right),\end{array}$$

(31)

where

$$\begin{array}{c}\hfill {\psi }_{p,m,\frac{1}{2},\frac{1}{2}}\left(x\right):=2^{-m}\sum _{i=1}^n{\left({⟨{\left|T_i\right|}^{\frac{p}{m}}x,x⟩}^{\frac{m}{2}}-{⟨{\left|T_i^{\ast }\right|}^{\frac{p}{m}}x,x⟩}^{\frac{m}{2}}\right)}^2.\end{array}$$

Proof. 

The desired result follows by taking $\alpha =\beta =\frac{1}{2}$ in (29). □

3. Upper and Lower Bounds for the Generalized Euclidean Operator Radius

In this section, we provide some lower and upper bounds for the product of the generalized Euclidean operator radius. In order to prove our results, we need to recall the following Hölder-type inequality:

$$\begin{array}{c}\hfill {\left(\sum _{j=1}^n{\left|x_j{y}_j\right|}^r\right)}^{\frac{1}{r}}\le {\left(\sum _{j=1}^n{\left|x_j\right|}^p\right)}^{\frac{1}{p}}{\left(\sum _{j=1}^n{\left|y_j\right|}^q\right)}^{\frac{1}{q}}\end{array}$$

(32)

for all complex numbers $x_j,y_j$$(1\le j\le n)$ and all $p,q,r\ge 1$ such that $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$.

Theorem 5.

Let $D_i,C_i,B_i,A_i\in \mathcal{B}\left(\mathcal{H}\right)$$(i=1,\cdots ,n)$, $r\ge 1$ and $p,q\ge 1$ with $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$. Then, we have

$$\begin{array}{cc}\hfill & \frac{1}{n^{2r-1}}{∥\sum _{i=1}^n{D}_i{C}_i{B}_i{A}_i∥}^{2r}\hfill \\ \hfill & \le {\omega }_p^r\left(A_1^{\ast }{\left|B_1\right|}^2{A}_1,\cdots ,A_n^{\ast }{\left|B_n\right|}^2{A}_n\right)w_q^r\left(D_1{\left|C_1^{\ast }\right|}^2{D}_1^{\ast },\cdots ,D_n{\left|C_n^{\ast }\right|}^2{D}_n^{\ast }\right)\hfill \\ \hfill & \le max\left\{\frac{r}{p},\frac{r}{q}\right\}∥\sum _{i=1}^n{\left(A_i^{\ast }{\left|B_i\right|}^2{A}_i\right)}^p+{\left(D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right)}^q∥-\underset{∥x∥=1}{inf}\lambda \left(x,y\right),\hfill \end{array}$$

(33)

where

$$\begin{array}{c}\hfill \lambda \left(x,y\right):=min\left\{\frac{r}{p},\frac{r}{q}\right\}{\left(\sqrt{\sum _{i=1}^n{⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,y⟩}^p}-\sqrt{\sum _{i=1}^n{⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,y⟩}^q}\right)}^2.\end{array}$$

Proof. 

Let $x,y\in \mathcal{H}$. Applying inequality (32) and the convexity of $t^{2r}$, we have

$$\begin{array}{cc}\hfill & \frac{1}{n^{2r-1}}{\left|⟨\left(\sum _{i=1}^n{D}_i{C}_i{B}_i{A}_i\right)x,y⟩\right|}^{2r}\hfill \\ \hfill & =\frac{1}{n^{2r-1}}{\left|\sum _{i=1}^n⟨\left(D_i{C}_i{B}_i{A}_i\right)x,y⟩\right|}^{2r}\hfill \\ \hfill & \le n{\left(\frac{1}{n}\sum _{i=1}^n\left|⟨\left(D_i{C}_i{B}_i{A}_i\right)x,y⟩\right|\right)}^{2r}\hfill \\ \hfill & \le \sum _{i=1}^n{\left|⟨\left(D_i{C}_i{B}_i{A}_i\right)x,y⟩\right|}^{2r}(\mathrm{by} \mathrm{Jensen}\text{’}\mathrm{s} \mathrm{inequality})\hfill \\ \hfill & \le \left(\sum _{i=1}^n{\left(⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,y⟩⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,y⟩\right)}^r\right)(\mathrm{by} (12\left)\right)\hfill \\ \hfill & \le {\left(\sum _{i=1}^n{⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,y⟩}^p\right)}^{\frac{r}{p}}{\left(\sum _{i=1}^n{⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,y⟩}^q\right)}^{\frac{r}{q}}(\mathrm{by} (32))\hfill \\ \hfill & \le \frac{r}{p}\sum _{i=1}^n{⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,y⟩}^p+\frac{r}{q}\sum _{i=1}^n{⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,y⟩}^q\mathrm{by} (10)\hfill \\ \hfill & -min\left\{\frac{r}{p},\frac{r}{q}\right\}{\left(\sqrt{\sum _{i=1}^n{⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,y⟩}^p}-\sqrt{\sum _{i=1}^n{⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,y⟩}^q}\right)}^2\hfill \\ \hfill & \le \frac{r}{p}\sum _{i=1}^n⟨{\left(A_i^{\ast }{\left|B_i\right|}^2{A}_i\right)}^{p}x,y⟩+\frac{r}{q}\sum _{i=1}^n⟨{\left(D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right)}^{q}x,y⟩(\mathrm{by} \mathrm{the} \mathrm{McCarthy} \mathrm{inequality})\hfill \\ \hfill & -min\left\{\frac{r}{p},\frac{r}{q}\right\}{\left(\sqrt{\sum _{i=1}^n{⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,y⟩}^p}-\sqrt{\sum _{i=1}^n{⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,y⟩}^q}\right)}^2\hfill \\ \hfill & \le max\left\{\frac{r}{p},\frac{r}{q}\right\}∥\sum _{i=1}^n{\left(A_i^{\ast }{\left|B_i\right|}^2{A}_i\right)}^p+{\left(D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }\right)}^q∥(\mathrm{properties} \mathrm{of} max)\hfill \\ \hfill & -min\left\{\frac{r}{p},\frac{r}{q}\right\}{\left(\sqrt{\sum _{i=1}^n{⟨A_i^{\ast }{\left|B_i\right|}^2{A}_{i}x,y⟩}^p}-\sqrt{\sum _{i=1}^n{⟨D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }x,y⟩}^q}\right)}^2.\hfill \end{array}$$

(34)

Taking the supremum over $x,y\in \mathcal{H}$ with $\parallel x\parallel =\parallel y\parallel =1$, the left- and right-hand sides follow immediately. The middle term of the inequality follows by (34), and thus, the desired result is obtained. □

Theorem 6.

Let $T_i\in \mathcal{B}\left(\mathcal{H}\right)$$(i=1,\cdots ,n)$, $r\ge 1$, $p,q\ge 1$ with $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ and $\alpha ,\beta \ge 0$ such that $\alpha +\beta \ge 1$. Then, we have

$$\begin{array}{cc}\hfill \frac{1}{n^{2r-1}}{∥\sum _{i=1}^n{T}_i{\left|T_i\right|}^{\alpha +\beta -1}∥}^{2r}& \le {\omega }_p^r\left({\left|T_1\right|}^{2\alpha },\cdots ,{\left|T_n\right|}^{2\alpha }\right)w_q^r\left({\left|T_1^{\ast }\right|}^{2\beta },\cdots ,{\left|T_n^{\ast }\right|}^{2\beta }\right)\hfill \\ \hfill & \le max\left\{\frac{r}{p},\frac{r}{q}\right\}∥\sum _{i=1}^n{\left|T_i\right|}^{2p\alpha }+{\left|T_i^{\ast }\right|}^{2q\beta }∥-\underset{∥x∥=∥y∥=1}{inf}\lambda \left(x,y\right),\hfill \end{array}$$

(35)

where

$$\begin{array}{c}\hfill \lambda \left(x,y\right):=min\left\{\frac{r}{p},\frac{r}{q}\right\}{\left(\sqrt{\sum _{i=1}^n{⟨{\left|T_i\right|}^{2\alpha }x,y⟩}^p}-\sqrt{\sum _{i=1}^n{⟨{\left|T_i^{\ast }\right|}^{2\beta }x,y⟩}^q}\right)}^2.\end{array}$$

Proof. 

Let $T_i=U_i\left|T_i\right|$ be the polar decomposition for ($i=1,\cdots ,n$). Setting $D_i=U_i$, $B=1_{\mathcal{H}}$, $C={\left|T_i\right|}^{\beta }$, and $A_i={\left|T_i\right|}^{\alpha }$ for all $\alpha ,\beta \ge 0$ such that $\alpha +\beta \ge 1$ in (28), then we have

$$\begin{array}{c}\hfill D_i{C}_i{B}_i{A}_i=U_i{\left|T_i\right|}^{\beta }{\left|T_i\right|}^{\alpha }=U_i\left|T_i\right|{\left|T_i\right|}^{\alpha +\beta -1}=T_i{\left|T_i\right|}^{\alpha +\beta -1}.\end{array}$$

Furthermore, we have $A_i^{\ast }{\left|B_i\right|}^2{A}_i={\left|T_i\right|}^{2\alpha }$ and $D_i{\left|C_i^{\ast }\right|}^2{D}_i^{\ast }=U_i{\left|T_i\right|}^{2\beta }{U}_i^{\ast }={\left|T_i^{\ast }\right|}^{2\beta }$ for all ($i=1,\cdots ,n$). □

Corollary 11.

Let $T_i\in \mathcal{B}\left(\mathcal{H}\right)$$(i=1,\cdots ,n)$, $r\ge 1$ and $p,q\ge 1$ with $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$. Then, we have

$$\begin{array}{cc}\hfill \frac{1}{n^{2r-1}}{∥\sum _{i=1}^n{T}_i∥}^{2r}& \le {\omega }_p^r\left(\left|T_1\right|,\cdots ,\left|T_n\right|\right)w_q^r\left(\left|T_1^{\ast }\right|,\cdots ,\left|T_n^{\ast }\right|\right)\hfill \\ \hfill & \le max\left\{\frac{r}{p},\frac{r}{q}\right\}∥\sum _{i=1}^n{\left|T_i\right|}^p+{\left|T_i^{\ast }\right|}^q∥-\underset{∥x∥=∥y∥=1}{inf}\lambda \left(x,y\right),\hfill \end{array}$$

(36)

where

$$\begin{array}{c}\hfill \lambda \left(x,y\right):=min\left\{\frac{r}{p},\frac{r}{q}\right\}{\left(\sqrt{\sum _{i=1}^n{⟨\left|T_i\right|x,y⟩}^p}-\sqrt{\sum _{i=1}^n{⟨\left|T_i^{\ast }\right|x,y⟩}^q}\right)}^2.\end{array}$$

Proof. 

It is enough to consider $\alpha =\beta =\frac{1}{2}$ in (35). □

Corollary 12.

Let $T_i\in \mathcal{B}\left(\mathcal{H}\right)$$(i=1,\cdots ,n)$, $\alpha ,\beta \ge 0$ such that $\alpha +\beta \ge 1$. Then, we have

$$\begin{array}{cc}\hfill \frac{1}{n}{∥\sum _{i=1}^n{T}_i{\left|T_i\right|}^{\alpha +\beta -1}∥}^2& \le w_{\mathrm{e}}\left({\left|T_1\right|}^{2\alpha },\cdots ,{\left|T_n\right|}^{2\alpha }\right)w_{\mathrm{e}}\left({\left|T_1^{\ast }\right|}^{2\beta },\cdots ,{\left|T_n^{\ast }\right|}^{2\beta }\right)\hfill \\ \hfill & \le \frac{1}{2}∥\sum _{i=1}^n{\left|T_i\right|}^{4\alpha }+{\left|T_i^{\ast }\right|}^{4\beta }∥-\underset{∥x∥=∥y∥=1}{inf}\lambda \left(x,y\right),\hfill \end{array}$$

(37)

where

$$\begin{array}{c}\hfill \lambda \left(x,y\right):=\frac{1}{2}{\left(\sqrt{\sum _{i=1}^n{⟨{\left|T_i\right|}^{2\alpha }x,y⟩}^2}-\sqrt{\sum _{i=1}^n{⟨{\left|T_i^{\ast }\right|}^{2\beta }x,y⟩}^2}\right)}^2.\end{array}$$

Proof. 

We obtain the result by taking $p=q=2$ and $r=1$ in (35). □

4. Concluding Remarks

In this paper, we have established numerous inequalities based on the generalized Euclidean operator radius, dealing with multivariable operators. Some of them extend and improve well-known numerical radius inequalities in the literature. We hope that the results and the diverse mathematical strategy will find applications in diverse mathematical areas dealing with operators. A possible direction is the fractal–wavelet analysis (see [39,40]).