{"id":"palomar/PALOMAR-2026-10-27-000021","url":"https://github.com/pragyaangaur/Real-Power-Cauchy-Schwarz/tree/8b7b39288079ad5b778b616efebd385816bf03d2","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-27-000021-v1.json","repo":"https://github.com/pragyaangaur/Real-Power-Cauchy-Schwarz","tags":["math.FA","math.CA","msc:26D15","msc:15A45","msc:15B48"],"score":0,"title":"A Cauchy-Schwarz inequality for real powers: proof, equality cases and weighted forms","trust":"high","domain":"math.FA","source":"palomar","authors":["Pragyaan Gaur"],"reviews":0,"subject":"math.FA","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-27","reactions":{},"formalized":true,"formalizes":[{"title":"A Cauchy-Schwarz inequality for real powers: proof, equality cases and weighted forms","identifier":"doi:10.5281/zenodo.23237507","relationship":"formalizes"}]}