On the $ρ$-numerical radius of rank-one operators and parametrized Buzano-type inequalities

Let $\mathcal{H}$ be a complex Hilbert space and $ρ>0$. We establish the explicit formula \[ ω_ρ(a\otimes b)=\frac{1}ρ\|a\| \|b\|+\left|1-\frac{1}ρ\right||\langle a,b\rangle|,\qquad a,b\in\mathcal{H}, \] for the $ρ$-numerical radius of rank-one operators, a unified expression interpolating b