A Sharp Norm Inequality and Buzano's Inequality via Determinants

We give a short linear-algebraic proof of the inequality $\|x\|_1\,\|x\|_\infty \le \frac{1+\sqrt{n}}{2}\,\|x\|_2^2,$ valid for every $x\in\mathbb{R}^n$. This inequality relates three fundamental norms on finite-dimensional spaces and has applications in optimization and numerical analysis. Our proof exploits the determinantal structure of a parametrized family of quadratic forms, and we show the constant $(1+\sqrt{n})/2$ is optimal. The inequality is a special case of Buzano's inequality, and the same method also proves Buzano's inequality for real vectors.