A Cauchy-Schwarz inequality for real powers: proof, equality cases and weighted forms
For a vector v with nonnegative entries and a real number p, write v^p for the vector of p-th powers of its entries. Johnston, Plosker, Torrance and Varona (arXiv:2507.10327, Conjecture 5.1) conjectured that ‖v^p‖ ‖w^p‖ - ⟨v^p, w^p⟩ ≤ ‖v‖^p ‖w‖^p - ⟨v, w⟩^p for every real p ≥ 2 and all vectors v, w in ℝⁿ with positive entries, with the Euclidean norm and inner product. They proved it for integer p and showed that it fails for 1 2 equality holds exactly when v and w are linearly dependent or each has at most one nonzero entry, and for p = 2 there is one further family supported on two…
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