Hadamard powers of Gram matrices and Generalized Cauchy-Schwarz inequalities

The following inequality for vectors $\mathbf{v},\mathbf{w}\in {\mathbb{R}}^n$ was considered in [Linear and Multilinear Algebra 74:1779–1798 (2026)]: $$\mathrm{\parallel }{\mathbf{v}}^p\mathrm{\parallel }\mathrm{\parallel }{\mathbf{w}}^p\mathrm{\parallel }-⟨{\mathbf{v}}^p,{\mathbf{w}}^p⟩\le \mathrm{\parallel }\mathbf{v}{\mathrm{\parallel }}^p\mathrm{\parallel }\mathbf{w}{\mathrm{\parallel }}^p-⟨\mathbf{v},\mathbf{w}{⟩}^p,$$ where ${\mathbf{v}}^p$ is the vector that is obtained by raising each entry of $\mathbf{v}$ to the power $p$. It was shown that this inequality holds for all positive integers $p$, and it was conjectured that it holds for all real $p\ge 2$ when the entries of $\mathbf{v}$ and $\mathbf{w}$ are positive. A formula for the maximum amount by which the inequality’s right-hand side can exceed its left-hand side for nonnegative unit vectors was also conjectured. We prove both of these conjectures, and we determine exactly when the inequality holds with equality. We also present a new generalization of this inequality to three or more vectors.