Generalized Euclidean operator radius inequalities of a pair of bounded linear operators

Let $\mathbb{B}(\mathscr{H})$ represent the $C^*$-algebra, which consists of all bounded linear operators on $\mathscr{H},$ and let $N ( .)$ be a norm on $\mathbb{B}(\mathscr{H})$. We define a norm $w_{(N,e)} (. , . )$ on $\mathbb{B}^2(\mathscr{H})$ by $$w_{(N,e)}(B,C)=\underset{|\lambda_1|^2+\lambda_2|^2\leq1}\sup \underset{\theta\in\mathbb{R}}\sup N\left(\Re \left(e^{i\theta}(\lambda_1B+\lambda_2C)\right)\right),$$ for every $B,C\in\mathbb{B}(\mathscr{H})$ and $\lambda_1,\lambda_2\in\mathbb{C}.$ We investigate basic properties of this norm and prove some bounds involving it. In particular, when $N( .)$ is the Hilbert-Schmidt norm, we prove some Hilbert-Schmidt Euclidean operator radius inequalities for a pair of bounded linear operators.