{"paper":{"title":"Majorization Inequalities from Logarithmic Convexity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Log-convexity in the indexing partition implies majorization inequalities for Macdonald polynomials, Jack polynomials, and Heckman-Opdam hypergeometric functions.","cross_cats":["math.RT"],"primary_cat":"math.CO","authors_text":"Colin McSwiggen, Siddhartha Sahi","submitted_at":"2026-05-12T19:30:23Z","abstract_excerpt":"Majorization inequalities for symmetric polynomials have interested mathematicians for centuries, from the AM-GM inequality for two variables going back at least to Euclid, through classical results of Newton, Muirhead and Gantmacher, to more recent extensions to Schur polynomials and zonal spherical functions. These have been established case by case, with no unified approach. Although it is known that majorization inequalities follow from symmetry and convexity in the indexing partition, the difficulty of proving convexity in specific cases has left a number of outstanding conjectures inacce"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Using log-convexity as a unifying principle, we prove new majorization inequalities for Macdonald polynomials, Jack polynomials and Heckman-Opdam hypergeometric functions, unifying existing results and resolving several open conjectures.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the specific functions (Macdonald, Jack, Heckman-Opdam) satisfy log-convexity in the indexing partition and that this property is preserved under the operations needed for the inductive arguments.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Log-convexity implies convexity and thus majorization inequalities for Macdonald polynomials, Jack polynomials, and Heckman-Opdam hypergeometric functions, unifying prior results and resolving open conjectures.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Log-convexity in the indexing partition implies majorization inequalities for Macdonald polynomials, Jack polynomials, and Heckman-Opdam hypergeometric functions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4c066870b2069443da8ae7701cea564e778339472acdcf2b79d575fa19fc8095"},"source":{"id":"2605.12680","kind":"arxiv","version":1},"verdict":{"id":"876c3509-5386-426d-8bf4-b9af1a2c51f9","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T20:16:47.166872Z","strongest_claim":"Using log-convexity as a unifying principle, we prove new majorization inequalities for Macdonald polynomials, Jack polynomials and Heckman-Opdam hypergeometric functions, unifying existing results and resolving several open conjectures.","one_line_summary":"Log-convexity implies convexity and thus majorization inequalities for Macdonald polynomials, Jack polynomials, and Heckman-Opdam hypergeometric functions, unifying prior results and resolving open conjectures.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the specific functions (Macdonald, Jack, Heckman-Opdam) satisfy log-convexity in the indexing partition and that this property is preserved under the operations needed for the inductive arguments.","pith_extraction_headline":"Log-convexity in the indexing partition implies majorization inequalities for Macdonald polynomials, Jack polynomials, and Heckman-Opdam hypergeometric functions."},"references":{"count":31,"sample":[{"doi":"","year":2018,"title":"R. Ait-Haddou and M.-L. Mazure,The fundamental blossoming inequality in Chebyshev spaces–I: Application to Schur functions, Foundations of Computational Mathematics18(2018), 135–158","work_id":"6a13ffa6-80cc-431b-b7bd-6262dd8c1a1d","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"B. Amri and K. Bedhiafi,A formula for the nonsymmetric Opdam’s hypergeometric function of typeA 2, Journal of Lie Theory27(2017), 309–335","work_id":"bd4df9e1-20b5-4a10-a364-27e1788efe4b","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"An introduction to Dunkl theory and its analytic aspects","work_id":"ee677f21-fd07-4aab-89b6-ac4ad6d9eb48","ref_index":3,"cited_arxiv_id":"1611.08213","is_internal_anchor":true},{"doi":"","year":2015,"title":"Alexei Borodin and Vadim Gorin,GeneralβJacobi corners process and the Gaussian free field, Communications on Pure and Applied Mathematics68(2015), no. 10, 1774–1844","work_id":"57b724b7-fba6-427e-a7ea-056a135ed29c","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Hong Chen, Apoorva Khare, and Siddhartha Sahi,Majorization via positivity of Jack and Macdonald polynomial differences, arXiv preprint arXiv:2509.19649 (2025),https://arxiv.org/abs/2509.19649","work_id":"43873faf-8472-4933-a10c-89d3183fddb6","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":31,"snapshot_sha256":"089ffe8a85b6d081e9ac8be6854be3023cef33328b0d155ad07c708e062f34e6","internal_anchors":3},"formal_canon":{"evidence_count":2,"snapshot_sha256":"0dd95d7d1a1c05a200340e14cceb3b6e5f915f212f07cc463b8691ec0c890480"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}