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For $0<p<\\infty$ and $f\\in\\mathcal{H}(\\mathbb{D})$, let $M_p^p(r,f)=\\int_0^{2\\pi}|f(re^{i\\theta})|^p \\frac{d\\theta}{2\\pi}$ and $M_\\infty(r,f) = \\sup_{|z|=r}|f(z)|$.\n  For $0<p<q\\leq \\infty$, Hardy and Littlewood proved the prevalent inequality $$M_q(r,f)\\le C(p,q)\\frac{M_p(\\rho,f)}{(\\rho-r)^{\\frac{1}{p}-\\frac{1}{q}}}$$ for $0\\leq r<\\rho\\leq 1$ and $f\\in\\mathcal{H}(\\mathbb{D})$.\n  In this paper, we obtain an improvement of this well-known inequality which is employed to c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.28080","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/2605.28080/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2605.28080","created_at":"2026-05-28T01:04:58.041681+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.28080v1","created_at":"2026-05-28T01:04:58.041681+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.28080","created_at":"2026-05-28T01:04:58.041681+00:00"},{"alias_kind":"pith_short_12","alias_value":"IXMGS5S7XQ6T","created_at":"2026-05-28T01:04:58.041681+00:00"},{"alias_kind":"pith_short_16","alias_value":"IXMGS5S7XQ6T43H2","created_at":"2026-05-28T01:04:58.041681+00:00"},{"alias_kind":"pith_short_8","alias_value":"IXMGS5S7","created_at":"2026-05-28T01:04:58.041681+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/IXMGS5S7XQ6T43H2QIDKDONNKO","json":"https://pith.science/pith/IXMGS5S7XQ6T43H2QIDKDONNKO.json","graph_json":"https://pith.science/api/pith-number/IXMGS5S7XQ6T43H2QIDKDONNKO/graph.json","events_json":"https://pith.science/api/pith-number/IXMGS5S7XQ6T43H2QIDKDONNKO/events.json","paper":"https://pith.science/paper/IXMGS5S7"},"agent_actions":{"view_html":"https://pith.science/pith/IXMGS5S7XQ6T43H2QIDKDONNKO","download_json":"https://pith.science/pith/IXMGS5S7XQ6T43H2QIDKDONNKO.json","view_paper":"https://pith.science/paper/IXMGS5S7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.28080&json=true","fetch_graph":"https://pith.science/api/pith-number/IXMGS5S7XQ6T43H2QIDKDONNKO/graph.json","fetch_events":"https://pith.science/api/pith-number/IXMGS5S7XQ6T43H2QIDKDONNKO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IXMGS5S7XQ6T43H2QIDKDONNKO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IXMGS5S7XQ6T43H2QIDKDONNKO/action/storage_attestation","attest_author":"https://pith.science/pith/IXMGS5S7XQ6T43H2QIDKDONNKO/action/author_attestation","sign_citation":"https://pith.science/pith/IXMGS5S7XQ6T43H2QIDKDONNKO/action/citation_signature","submit_replication":"https://pith.science/pith/IXMGS5S7XQ6T43H2QIDKDONNKO/action/replication_record"}},"created_at":"2026-05-28T01:04:58.041681+00:00","updated_at":"2026-05-28T01:04:58.041681+00:00"}