$q$-numerical radius of rank-one operators and the generalized Buzano inequality

Here, we study the $q$-numerical radius of rank-one operators on a Hilbert space $\mathcal{H}$. More precisely, for $q\in [0,1]$ and $a,b\in \mathcal{H}$, we establish the formula $${\omega }_q(a\otimes b)=\frac{1}{2}(\mathrm{\parallel }a\mathrm{\parallel }\mathrm{\parallel }b\mathrm{\parallel }+q\mathrm{\mid }⟨a,b⟩\mathrm{\mid }+\sqrt{1-q^2}\sqrt{\mathrm{\parallel }a{\mathrm{\parallel }}^2\mathrm{\parallel }b{\mathrm{\parallel }}^2-\mathrm{\mid }⟨a,b⟩{\mathrm{\mid }}^2}),$$ which represents a generalization of the well-known formula for the numerical radius of a rank-one operator in a Hilbert space, obtained by setting $q=1$. As a corollary, we also derive a generalization of the classical Buzano inequality.