Generalizing the Cauchy-Schwarz inequality: Hadamard powers and tensor products

We explore and generalize a Cauchy-Schwarz-type inequality originally proved in [Electronic Journal of Linear Algebra 35, 156-180 (2019)]: $\mathrm{\parallel }{\mathbf{v}}^2\mathrm{\parallel }\mathrm{\parallel }{\mathbf{w}}^2\mathrm{\parallel }-⟨{\mathbf{v}}^2,{\mathbf{w}}^2⟩\le \mathrm{\parallel }\mathbf{v}{\mathrm{\parallel }}^2\mathrm{\parallel }\mathbf{w}{\mathrm{\parallel }}^2-⟨\mathbf{v},\mathbf{w}{⟩}^2$ for all $\mathbf{v},\mathbf{w}\in {\mathbb{R}}^n$. We present three new proofs of this inequality that better illustrate “why” it is true and generalize it in several different ways: we generalize from vectors to matrices, we explore which exponents other than 2 result in the inequality holding, and we derive a version of the inequality involving three or more vectors.