Estimations of Euclidean operator radius

We develop several Euclidean operator radius bounds for the product of two $d$-tuple operators using positivity criteria of a $2\times 2$ block matrix whose entries are $d$-tuple operators. From these bounds, by using the polar decomposition of operators, we obtain Euclidean operator radius bounds for $d$-tuple operators. Among many other interesting bounds, it is shown that \begin{eqnarray*} w_e($\mathbf{A}$) &\leq&\frac1{\sqrt2} $\mathbf{A}$|^{1/2}\sqrt{\left|\sum_{k=1}^{d} (|A_k|+|A_k^|)\right|}, \end{eqnarray} where $w_e(\mathbf{A})$ and $\mathrm{\parallel }\mathbf{A}\mathrm{\parallel }$ are the Euclidean operator radius and the Euclidean operator norm, respectively, of a $d$-tuple operator $\mathbf{A}=(A_1,A_2,\dots ,A_d)\mathrm{.}$ Further, we develop an upper bound for the Euclidean operator radius of $n\times n$ operator matrix whose entries are $d$-tuple operators. In particular, it is proved that if ${\left[\begin{array}{c}{\mathbf{A}}_{\mathbf{i}\mathbf{j}}\end{array}\right]}_{n\times n}$ is an $n\times n$ operator matrix then $$w_e\left({\left[\begin{array}{c}{\mathbf{A}}_{\mathbf{i}\mathbf{j}}\end{array}\right]}_{n\times n}\right)\le w\left({\left[\begin{array}{c}{a}_{ij}\end{array}\right]}_{n\times n}\right),$$ where each ${\mathbf{A}}_{\mathbf{i}\mathbf{j}}$ is a $d$-tuple operator, $1\le i,j\le n$, $a_{ij}=w_e({\mathbf{A}}_{\mathbf{i}\mathbf{j}})\mathit{\text{if}}\mathit{\text{i=j}}$, a_{ij}= \sqrt{w_e\left(|\mathbf{A_{ji}|}+|\mathbf{A_{ij}^*}|\right)w_e\left(|\mathbf{A_{ij}|}+|\mathbf{A_{ji}^*}|\right)}\,\textit{ ifi<j}, and a_{ij}= 0\,\textit{ ifi>j}. Other related applications are also discussed.