This paper investigates power vector inequalities for bounded linear operator pairs $(B,C)$ in a Hilbert space $\mathcal{H}$ . By leveraging vector inequalities derived by the second author for inner products and norms, we establish various power vector inequalities for operator pairs in Hilbert spaces. Specifically, we analyze cases where $(B,C)$ corresponds to $\left (A,A^{\ast }\right )$ or $\left (\mathfrak{R}(A),\mathfrak{I}(A)\right )$ , where $A^{\ast}$ , $\mathfrak{R}(A)$ , and $\mathfrak{I}(A)$ denote the adjoint, real, and imaginary parts of A. This analysis leads to the derivation of vector, norm, and numerical radius inequalities for single operators. Additionally, we derive power inequalities for the s-r-norm and s-r-numerical radius ( $r\geq 1$ ) of the operator pair $(B,C)$ , which generalize the Euclidean norm and Euclidean numerical radius for operator pairs. Finally, we apply these results to derive corresponding inequalities for a single bounded linear operator A on $\mathcal{H}$ .
