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We obtain some properties of the set function $F:\\Omega\\mapsto \\R^+$ defined by $$ F(\\Omega)=\\frac{T(\\Omega)\\lambda_1(\\Omega)}{|\\Omega|} ,$$ where $T(\\Omega)$ and $\\lambda_1(\\Omega)$ are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical P\\'olya bound $F(\\Omega)\\le 1,$ and show that $$F(\\Omega)\\le 1- \\nu_m T(\\Omega)|\\Omega|^{-1-\\frac2m},$$ where $\\nu_m$ depends only on $m$. For any $m=2,3,\\dots$ and $\\epsilon\\in (0,1)$ we construct an open set $"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1602.04618","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-02-15T10:46:05Z","cross_cats_sorted":[],"title_canon_sha256":"23617ebbbcf596c3a3bb6e8a08e8ec233d1dfd67103302ebd70024b0aaa08897","abstract_canon_sha256":"ccbefe8e9116c437b664c2972e98c85ab00c82fe1feb93aa78688f26f4a4314b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:47:38.293531Z","signature_b64":"iV1janzkoUh5/Tc+lQCnGqRUkShjtVTdLK9usQnKIUQhv2YyF5nZ7Crm6lepN2hAXCdVhc6TR03rFX8gc01iCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1a648fcfacfb3f99f6d95f59648ac1aef739a92c7aa4870205d5902a821b428a","last_reissued_at":"2026-05-18T00:47:38.292944Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:47:38.292944Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Polya's inequality for torsional rigidity and first Dirichlet eigenvalue","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"C. Nitsch, C. Trombetti, M. van den Berg, V. Ferone","submitted_at":"2016-02-15T10:46:05Z","abstract_excerpt":"Let $\\Omega$ be an open set in Euclidean space with finite Lebesgue measure $|\\Omega|$. We obtain some properties of the set function $F:\\Omega\\mapsto \\R^+$ defined by $$ F(\\Omega)=\\frac{T(\\Omega)\\lambda_1(\\Omega)}{|\\Omega|} ,$$ where $T(\\Omega)$ and $\\lambda_1(\\Omega)$ are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical P\\'olya bound $F(\\Omega)\\le 1,$ and show that $$F(\\Omega)\\le 1- \\nu_m T(\\Omega)|\\Omega|^{-1-\\frac2m},$$ where $\\nu_m$ depends only on $m$. For any $m=2,3,\\dots$ and $\\epsilon\\in (0,1)$ we construct an open set $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.04618","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1602.04618","created_at":"2026-05-18T00:47:38.293045+00:00"},{"alias_kind":"arxiv_version","alias_value":"1602.04618v5","created_at":"2026-05-18T00:47:38.293045+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.04618","created_at":"2026-05-18T00:47:38.293045+00:00"},{"alias_kind":"pith_short_12","alias_value":"DJSI7T5M7M7Z","created_at":"2026-05-18T12:30:12.583610+00:00"},{"alias_kind":"pith_short_16","alias_value":"DJSI7T5M7M7ZT5WZ","created_at":"2026-05-18T12:30:12.583610+00:00"},{"alias_kind":"pith_short_8","alias_value":"DJSI7T5M","created_at":"2026-05-18T12:30:12.583610+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/DJSI7T5M7M7ZT5WZL5MWJCWBV3","json":"https://pith.science/pith/DJSI7T5M7M7ZT5WZL5MWJCWBV3.json","graph_json":"https://pith.science/api/pith-number/DJSI7T5M7M7ZT5WZL5MWJCWBV3/graph.json","events_json":"https://pith.science/api/pith-number/DJSI7T5M7M7ZT5WZL5MWJCWBV3/events.json","paper":"https://pith.science/paper/DJSI7T5M"},"agent_actions":{"view_html":"https://pith.science/pith/DJSI7T5M7M7ZT5WZL5MWJCWBV3","download_json":"https://pith.science/pith/DJSI7T5M7M7ZT5WZL5MWJCWBV3.json","view_paper":"https://pith.science/paper/DJSI7T5M","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1602.04618&json=true","fetch_graph":"https://pith.science/api/pith-number/DJSI7T5M7M7ZT5WZL5MWJCWBV3/graph.json","fetch_events":"https://pith.science/api/pith-number/DJSI7T5M7M7ZT5WZL5MWJCWBV3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DJSI7T5M7M7ZT5WZL5MWJCWBV3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DJSI7T5M7M7ZT5WZL5MWJCWBV3/action/storage_attestation","attest_author":"https://pith.science/pith/DJSI7T5M7M7ZT5WZL5MWJCWBV3/action/author_attestation","sign_citation":"https://pith.science/pith/DJSI7T5M7M7ZT5WZL5MWJCWBV3/action/citation_signature","submit_replication":"https://pith.science/pith/DJSI7T5M7M7ZT5WZL5MWJCWBV3/action/replication_record"}},"created_at":"2026-05-18T00:47:38.293045+00:00","updated_at":"2026-05-18T00:47:38.293045+00:00"}