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Here $G(a,b)$, $L(a,b)$, $AG(a,b)$ and $A(a,b)$ are the geometric, logarithmic, arithmetic-geometric and arithmetic means of $a$ and $b$, respectively."},"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":"1209.3350","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-09-15T03:20:24Z","cross_cats_sorted":[],"title_canon_sha256":"cf0f02cff4545bb6bdeff1d50786668581d16b0c207d1244933d0abcbfe1f325","abstract_canon_sha256":"ad15225b953a7925ad6ed60ed474bf863bdc21b1feabd0df7cefb06851e4fdde"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:42.445204Z","signature_b64":"BP65+xkX2PIC5z/x68b0Iv7WTx264+hvVFF6J4oFe7ayig3cOlU1XMtCkqftQVHvPFISskfSWGaZbU0imjjYDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8736d6aceb6118cccc1e3ec7d88f9636f4ab097aa396d329e3a44f011447b6f9","last_reissued_at":"2026-05-18T03:41:42.444437Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:42.444437Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp two parameter bounds for logarithmic and arithmetic-geometric means","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Miao-Kun Wang, Xiao-Yan Ma, Ye-Fang Qiu, Yu-Ming Chu","submitted_at":"2012-09-15T03:20:24Z","abstract_excerpt":"For fixed $s\\geq 1$ and $t_{1},t_{2}\\in(0,1/2)$ we prove that the inequalities $G^{s}(t_{1}a+(1-t_{1})b,t_{1}b+(1-t_{1})a)A^{1-s}(a,b)>AG(a,b)$ and $G^{s}(t_{2}a+(1-t_{2})b,t_{2}b+(1-t_{2})a)A^{1-s}(a,b)>L(a,b)$ hold for all $a,b>0$ with $a\\neq b$ if and only if $t_{1}\\geq 1/2-\\sqrt{2s}/(4s)$ and $t_{2}\\geq 1/2-\\sqrt{6s}/(6s)$. Here $G(a,b)$, $L(a,b)$, $AG(a,b)$ and $A(a,b)$ are the geometric, logarithmic, arithmetic-geometric and arithmetic means of $a$ and $b$, respectively."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3350","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":""},"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":"1209.3350","created_at":"2026-05-18T03:41:42.444557+00:00"},{"alias_kind":"arxiv_version","alias_value":"1209.3350v1","created_at":"2026-05-18T03:41:42.444557+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.3350","created_at":"2026-05-18T03:41:42.444557+00:00"},{"alias_kind":"pith_short_12","alias_value":"Q43NNLHLMEMM","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_16","alias_value":"Q43NNLHLMEMMZTA6","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_8","alias_value":"Q43NNLHL","created_at":"2026-05-18T12:27:18.751474+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/Q43NNLHLMEMMZTA6H3D5RD4WG3","json":"https://pith.science/pith/Q43NNLHLMEMMZTA6H3D5RD4WG3.json","graph_json":"https://pith.science/api/pith-number/Q43NNLHLMEMMZTA6H3D5RD4WG3/graph.json","events_json":"https://pith.science/api/pith-number/Q43NNLHLMEMMZTA6H3D5RD4WG3/events.json","paper":"https://pith.science/paper/Q43NNLHL"},"agent_actions":{"view_html":"https://pith.science/pith/Q43NNLHLMEMMZTA6H3D5RD4WG3","download_json":"https://pith.science/pith/Q43NNLHLMEMMZTA6H3D5RD4WG3.json","view_paper":"https://pith.science/paper/Q43NNLHL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1209.3350&json=true","fetch_graph":"https://pith.science/api/pith-number/Q43NNLHLMEMMZTA6H3D5RD4WG3/graph.json","fetch_events":"https://pith.science/api/pith-number/Q43NNLHLMEMMZTA6H3D5RD4WG3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Q43NNLHLMEMMZTA6H3D5RD4WG3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Q43NNLHLMEMMZTA6H3D5RD4WG3/action/storage_attestation","attest_author":"https://pith.science/pith/Q43NNLHLMEMMZTA6H3D5RD4WG3/action/author_attestation","sign_citation":"https://pith.science/pith/Q43NNLHLMEMMZTA6H3D5RD4WG3/action/citation_signature","submit_replication":"https://pith.science/pith/Q43NNLHLMEMMZTA6H3D5RD4WG3/action/replication_record"}},"created_at":"2026-05-18T03:41:42.444557+00:00","updated_at":"2026-05-18T03:41:42.444557+00:00"}