{"paper":{"title":"A Sharp Inequality of Hardy-Littlewood Type Via Derivatives","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.FA","authors_text":"Hui Dan, Kunyu Guo, Yi Wang","submitted_at":"2019-08-04T11:46:14Z","abstract_excerpt":"In this paper we consider a generalized version of Carleman's inequality. An equivalent version of it states that $\\|f\\|_{A_\\alpha^{2\\alpha}}\\leq\\|f\\|_{H^2}$, where $f$ is a holomorphic function and $\\alpha>1$. If the norms $\\|f\\|_{A_\\alpha^{2\\alpha}}$ are decreasing in $\\alpha$, then the inequality holds for $f$. For a dense set of functions, we calculate the derivative of the norms $\\|f\\|_{A_\\alpha^{2\\alpha}}$ in $\\alpha$ and give sufficient conditions for this derivative to be non-positive. As an application, we prove the inequality for linear combinations of two reproducing kernels. Some n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.01320","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.01320/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}