{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:SR4C57KMC6KG2K2AJU6BSFJHDL","short_pith_number":"pith:SR4C57KM","canonical_record":{"source":{"id":"1402.4561","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-02-19T06:03:43Z","cross_cats_sorted":[],"title_canon_sha256":"9c2117f1e4f42ec7c6e2a69a7240ed9ff55f37daea31dd05cfc7d4238e4ea5a3","abstract_canon_sha256":"0a3579f468da62a93070a37830441031aeda6bd81a0565c52de4272aca850410"},"schema_version":"1.0"},"canonical_sha256":"94782efd4c17946d2b404d3c1915271ae666b21d4ade05eaf525b5eba5569314","source":{"kind":"arxiv","id":"1402.4561","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.4561","created_at":"2026-05-18T01:11:43Z"},{"alias_kind":"arxiv_version","alias_value":"1402.4561v1","created_at":"2026-05-18T01:11:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.4561","created_at":"2026-05-18T01:11:43Z"},{"alias_kind":"pith_short_12","alias_value":"SR4C57KMC6KG","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_16","alias_value":"SR4C57KMC6KG2K2A","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_8","alias_value":"SR4C57KM","created_at":"2026-05-18T12:28:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:SR4C57KMC6KG2K2AJU6BSFJHDL","target":"record","payload":{"canonical_record":{"source":{"id":"1402.4561","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-02-19T06:03:43Z","cross_cats_sorted":[],"title_canon_sha256":"9c2117f1e4f42ec7c6e2a69a7240ed9ff55f37daea31dd05cfc7d4238e4ea5a3","abstract_canon_sha256":"0a3579f468da62a93070a37830441031aeda6bd81a0565c52de4272aca850410"},"schema_version":"1.0"},"canonical_sha256":"94782efd4c17946d2b404d3c1915271ae666b21d4ade05eaf525b5eba5569314","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:43.990653Z","signature_b64":"g1if28ZQ99NYaf06H83E+8JXuIaGQEqhiIHDJwsUjfXyJzI9Nn25JYJuvnDI5Rt4o6aw9IffvvlDNfOlMFSRBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"94782efd4c17946d2b404d3c1915271ae666b21d4ade05eaf525b5eba5569314","last_reissued_at":"2026-05-18T01:11:43.990128Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:43.990128Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1402.4561","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:11:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uU0xZ0t1oTq4oox3bb3U89ePNVP/R2Fx25ytYO3b78S9CRRrRN9Fhox1eiwY2M2R0e2StTY4MGAlivBRFNSdDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T16:44:52.793189Z"},"content_sha256":"bc5105c5cb512b9f6eed12f99f2fac55dd9893eb085e99188ee05135631074c9","schema_version":"1.0","event_id":"sha256:bc5105c5cb512b9f6eed12f99f2fac55dd9893eb085e99188ee05135631074c9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:SR4C57KMC6KG2K2AJU6BSFJHDL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bounds for the combination of Toader mean and the arithmetic mean in terms of the contraharmonic mean","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Feng Qi, Wei-Dong Jiang","submitted_at":"2014-02-19T06:03:43Z","abstract_excerpt":"In the paper, the authors find the greatest value $\\lambda$ and the least value $\\mu $ such that the double inequality \\begin{multline*} C(\\lambda a+(1-\\lambda)b,\\lambda b+(1-\\lambda )a)<\\alpha A(a,b)+(1-\\alpha)T(a,b)\\\\ < C(\\mu a+(1-\\mu)b,\\mu b+(1-\\mu )a) \\end{multline*} holds for all $\\alpha\\in(0,1)$ and $a,b>0$ with $a\\ne b$, where $$ C(a,b)=\\frac{a^{2}+b^{2}}{a+b},\\quad A(a,b)=\\frac{a+b}2, $$ and $$ T(a,b)=\\frac{2}{\\pi}\\int_{0}^{{\\pi}/{2}}\\sqrt{a^2\\cos^2\\theta+b^2\\sin^2\\theta}\\,d\\theta$$ denote respectively the contraharmonic, arithmetic, and Toader means of two positive numbers $a$ and $b$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.4561","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:11:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YnslNW2Z44MEV/vkLYvyDyN6wAW6OwcbNNBkVo9u7tiow7HplOsc0xJfdcwOEEUPFte/uO73EUODRhcdUlbbAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T16:44:52.793840Z"},"content_sha256":"8957ba1cd65a7a4f75b8c8534fc36841be2ec56237efa5baf5c51fed8abb2f81","schema_version":"1.0","event_id":"sha256:8957ba1cd65a7a4f75b8c8534fc36841be2ec56237efa5baf5c51fed8abb2f81"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SR4C57KMC6KG2K2AJU6BSFJHDL/bundle.json","state_url":"https://pith.science/pith/SR4C57KMC6KG2K2AJU6BSFJHDL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SR4C57KMC6KG2K2AJU6BSFJHDL/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-08T16:44:52Z","links":{"resolver":"https://pith.science/pith/SR4C57KMC6KG2K2AJU6BSFJHDL","bundle":"https://pith.science/pith/SR4C57KMC6KG2K2AJU6BSFJHDL/bundle.json","state":"https://pith.science/pith/SR4C57KMC6KG2K2AJU6BSFJHDL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SR4C57KMC6KG2K2AJU6BSFJHDL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:SR4C57KMC6KG2K2AJU6BSFJHDL","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"0a3579f468da62a93070a37830441031aeda6bd81a0565c52de4272aca850410","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-02-19T06:03:43Z","title_canon_sha256":"9c2117f1e4f42ec7c6e2a69a7240ed9ff55f37daea31dd05cfc7d4238e4ea5a3"},"schema_version":"1.0","source":{"id":"1402.4561","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.4561","created_at":"2026-05-18T01:11:43Z"},{"alias_kind":"arxiv_version","alias_value":"1402.4561v1","created_at":"2026-05-18T01:11:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.4561","created_at":"2026-05-18T01:11:43Z"},{"alias_kind":"pith_short_12","alias_value":"SR4C57KMC6KG","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_16","alias_value":"SR4C57KMC6KG2K2A","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_8","alias_value":"SR4C57KM","created_at":"2026-05-18T12:28:49Z"}],"graph_snapshots":[{"event_id":"sha256:8957ba1cd65a7a4f75b8c8534fc36841be2ec56237efa5baf5c51fed8abb2f81","target":"graph","created_at":"2026-05-18T01:11:43Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In the paper, the authors find the greatest value $\\lambda$ and the least value $\\mu $ such that the double inequality \\begin{multline*} C(\\lambda a+(1-\\lambda)b,\\lambda b+(1-\\lambda )a)<\\alpha A(a,b)+(1-\\alpha)T(a,b)\\\\ < C(\\mu a+(1-\\mu)b,\\mu b+(1-\\mu )a) \\end{multline*} holds for all $\\alpha\\in(0,1)$ and $a,b>0$ with $a\\ne b$, where $$ C(a,b)=\\frac{a^{2}+b^{2}}{a+b},\\quad A(a,b)=\\frac{a+b}2, $$ and $$ T(a,b)=\\frac{2}{\\pi}\\int_{0}^{{\\pi}/{2}}\\sqrt{a^2\\cos^2\\theta+b^2\\sin^2\\theta}\\,d\\theta$$ denote respectively the contraharmonic, arithmetic, and Toader means of two positive numbers $a$ and $b$","authors_text":"Feng Qi, Wei-Dong Jiang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-02-19T06:03:43Z","title":"Bounds for the combination of Toader mean and the arithmetic mean in terms of the contraharmonic mean"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.4561","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:bc5105c5cb512b9f6eed12f99f2fac55dd9893eb085e99188ee05135631074c9","target":"record","created_at":"2026-05-18T01:11:43Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"0a3579f468da62a93070a37830441031aeda6bd81a0565c52de4272aca850410","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-02-19T06:03:43Z","title_canon_sha256":"9c2117f1e4f42ec7c6e2a69a7240ed9ff55f37daea31dd05cfc7d4238e4ea5a3"},"schema_version":"1.0","source":{"id":"1402.4561","kind":"arxiv","version":1}},"canonical_sha256":"94782efd4c17946d2b404d3c1915271ae666b21d4ade05eaf525b5eba5569314","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"94782efd4c17946d2b404d3c1915271ae666b21d4ade05eaf525b5eba5569314","first_computed_at":"2026-05-18T01:11:43.990128Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:11:43.990128Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"g1if28ZQ99NYaf06H83E+8JXuIaGQEqhiIHDJwsUjfXyJzI9Nn25JYJuvnDI5Rt4o6aw9IffvvlDNfOlMFSRBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:11:43.990653Z","signed_message":"canonical_sha256_bytes"},"source_id":"1402.4561","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bc5105c5cb512b9f6eed12f99f2fac55dd9893eb085e99188ee05135631074c9","sha256:8957ba1cd65a7a4f75b8c8534fc36841be2ec56237efa5baf5c51fed8abb2f81"],"state_sha256":"f2a3c97d631ae74078d99a2588f7420081f173296feb0af11dd174c5bc08a292"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5MTvSTFUbj10auN8LkcDzfvnqxpuBbDrVoaxpjZ1KWziwjAlakmcIKiMGYB0DOA4lFjW9/P1TtOHXNlU7oaSCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T16:44:52.797549Z","bundle_sha256":"901544873f0d34b8d3a3314712274ab208aac0974a7aa2588998a74006bffc8b"}}