The article examines inequalities for norms and numerical radii of bounded linear operators on complex Hilbert spaces. It focuses on scenarios where three operators are involved, with one being positive, and investigates their sums or products. Some of our findings extend existing inequalities established in the literature.

For $\nu \in [0,1]$, $p\ge 1$ and $A,B\in \mathcal{B}\left(H\right)$, we define the $p$-arithmetic-mean (A-M)-norm for the pair of operators $\left(A,B\right)$ by

$${\Vert \left(A,B\right)\Vert }_{p,\nu }:=\underset{\Vert x\Vert =1}{sup}{\left(\left(1-\nu \right){\Vert Ax\Vert }^p+\nu {\Vert Bx\Vert }^p\right)}^{1/p}.$$

In this paper, we obtain several lower and upper bounds for this norm. Some inequalities for the numerical radius of the off-diagonal operator matrix are given. In the case when $\left(A,B\right)=\left(T,T^{\ast }\right)$ and $\left(A,B\right)=\left(\mathrm{R}\mathrm{e}T,\mathrm{I}\mathrm{m}T\right)$, where $\mathrm{R}\mathrm{e}T:=\frac{T+T^{\ast }}{2}$ is the real part of $T$ and $\mathrm{I}\mathrm{m}T:=\frac{T-T^{\ast }}{2\text{i}}$ is the imaginary part of $T$, respectively, some inequalities for one operator are also provided.