{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:KSZRGIFBI6EJLYUNXQBWGR5FRG","short_pith_number":"pith:KSZRGIFB","canonical_record":{"source":{"id":"1112.0686","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2011-12-03T19:45:10Z","cross_cats_sorted":[],"title_canon_sha256":"6c004c7bc938255fb31431448f6f54260d4a315d3e8a6509fd7e22d9426e664e","abstract_canon_sha256":"c5232a9c5871f92be0b70396e1f857c1787ac3103f04e801d9f1d3f25c75ce27"},"schema_version":"1.0"},"canonical_sha256":"54b31320a1478895e28dbc036347a589890110c9e689808881c53f2dc7243a39","source":{"kind":"arxiv","id":"1112.0686","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.0686","created_at":"2026-05-18T04:07:05Z"},{"alias_kind":"arxiv_version","alias_value":"1112.0686v1","created_at":"2026-05-18T04:07:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.0686","created_at":"2026-05-18T04:07:05Z"},{"alias_kind":"pith_short_12","alias_value":"KSZRGIFBI6EJ","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"KSZRGIFBI6EJLYUN","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"KSZRGIFB","created_at":"2026-05-18T12:26:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:KSZRGIFBI6EJLYUNXQBWGR5FRG","target":"record","payload":{"canonical_record":{"source":{"id":"1112.0686","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2011-12-03T19:45:10Z","cross_cats_sorted":[],"title_canon_sha256":"6c004c7bc938255fb31431448f6f54260d4a315d3e8a6509fd7e22d9426e664e","abstract_canon_sha256":"c5232a9c5871f92be0b70396e1f857c1787ac3103f04e801d9f1d3f25c75ce27"},"schema_version":"1.0"},"canonical_sha256":"54b31320a1478895e28dbc036347a589890110c9e689808881c53f2dc7243a39","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:07:05.952075Z","signature_b64":"M/mO/QWC5GcC3DEf6DkH9q80Z6GVx2Z31byaFpkTDEFWxNGvuXJs9Z7uR/4Xj8x5Xb2fxlmr0pLIo6a/iQhhAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"54b31320a1478895e28dbc036347a589890110c9e689808881c53f2dc7243a39","last_reissued_at":"2026-05-18T04:07:05.951624Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:07:05.951624Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1112.0686","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":"2026-05-18T04:07:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A+8Efd6N65jjwYaMOWo6JyCqkfZ0yP11HZrKv3NPa4+9ge2aWUVoaFWBF0KRaXboA6jTm90IIDL19P3kNUL0AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T06:41:34.186816Z"},"content_sha256":"afe01b657c969332445cf859b9e9701dd30e1b038cffaa04d62c913a2effe54b","schema_version":"1.0","event_id":"sha256:afe01b657c969332445cf859b9e9701dd30e1b038cffaa04d62c913a2effe54b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:KSZRGIFBI6EJLYUNXQBWGR5FRG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On harmonic combination of univalent functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"M. Obradovi\\'c, S.Ponnusamy","submitted_at":"2011-12-03T19:45:10Z","abstract_excerpt":"Let ${\\mathcal S}$ be the class of all functions $f$ that are analytic and univalent in the unit disk $\\ID$ with the normalization $f(0)=f'(0)-1=0$. Let $\\mathcal{U} (\\lambda)$ denote the set of all $f\\in {\\mathcal S}$ satisfying the condition $$|f'(z)(\\frac{z}{f(z)})^{2}-1| <\\lambda ~for $z\\in \\ID$, $$ for some $\\lambda \\in (0,1]$. In this paper, among other things, we study a \"harmonic mean\" of two univalent analytic functions. More precisely, we discuss the properties of the class of functions $F$ of the form $$\\frac{z}{F(z)}=1/2(\\frac{z}{f(z)}+\\frac{z}{g(z)}), $$ where $f,g\\in \\mathcal{S}$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.0686","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":"2026-05-18T04:07:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9cMcuzB59vnS0wxpoLAWuCtEZIBIyTWUCRS6hUljTkyiC2I7tW5EDiKxjXH7Fr4hWn3FT+NASRndEEe43JSUCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T06:41:34.187163Z"},"content_sha256":"960dc6f95a58675df55ea3457782a5f6656aac132e79dd8b8c76229d55a2c0e3","schema_version":"1.0","event_id":"sha256:960dc6f95a58675df55ea3457782a5f6656aac132e79dd8b8c76229d55a2c0e3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KSZRGIFBI6EJLYUNXQBWGR5FRG/bundle.json","state_url":"https://pith.science/pith/KSZRGIFBI6EJLYUNXQBWGR5FRG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KSZRGIFBI6EJLYUNXQBWGR5FRG/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":"2026-06-27T06:41:34Z","links":{"resolver":"https://pith.science/pith/KSZRGIFBI6EJLYUNXQBWGR5FRG","bundle":"https://pith.science/pith/KSZRGIFBI6EJLYUNXQBWGR5FRG/bundle.json","state":"https://pith.science/pith/KSZRGIFBI6EJLYUNXQBWGR5FRG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KSZRGIFBI6EJLYUNXQBWGR5FRG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:KSZRGIFBI6EJLYUNXQBWGR5FRG","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":"c5232a9c5871f92be0b70396e1f857c1787ac3103f04e801d9f1d3f25c75ce27","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2011-12-03T19:45:10Z","title_canon_sha256":"6c004c7bc938255fb31431448f6f54260d4a315d3e8a6509fd7e22d9426e664e"},"schema_version":"1.0","source":{"id":"1112.0686","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.0686","created_at":"2026-05-18T04:07:05Z"},{"alias_kind":"arxiv_version","alias_value":"1112.0686v1","created_at":"2026-05-18T04:07:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.0686","created_at":"2026-05-18T04:07:05Z"},{"alias_kind":"pith_short_12","alias_value":"KSZRGIFBI6EJ","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"KSZRGIFBI6EJLYUN","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"KSZRGIFB","created_at":"2026-05-18T12:26:34Z"}],"graph_snapshots":[{"event_id":"sha256:960dc6f95a58675df55ea3457782a5f6656aac132e79dd8b8c76229d55a2c0e3","target":"graph","created_at":"2026-05-18T04:07:05Z","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":"Let ${\\mathcal S}$ be the class of all functions $f$ that are analytic and univalent in the unit disk $\\ID$ with the normalization $f(0)=f'(0)-1=0$. Let $\\mathcal{U} (\\lambda)$ denote the set of all $f\\in {\\mathcal S}$ satisfying the condition $$|f'(z)(\\frac{z}{f(z)})^{2}-1| <\\lambda ~for $z\\in \\ID$, $$ for some $\\lambda \\in (0,1]$. In this paper, among other things, we study a \"harmonic mean\" of two univalent analytic functions. More precisely, we discuss the properties of the class of functions $F$ of the form $$\\frac{z}{F(z)}=1/2(\\frac{z}{f(z)}+\\frac{z}{g(z)}), $$ where $f,g\\in \\mathcal{S}$","authors_text":"M. Obradovi\\'c, S.Ponnusamy","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2011-12-03T19:45:10Z","title":"On harmonic combination of univalent functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.0686","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:afe01b657c969332445cf859b9e9701dd30e1b038cffaa04d62c913a2effe54b","target":"record","created_at":"2026-05-18T04:07:05Z","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":"c5232a9c5871f92be0b70396e1f857c1787ac3103f04e801d9f1d3f25c75ce27","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2011-12-03T19:45:10Z","title_canon_sha256":"6c004c7bc938255fb31431448f6f54260d4a315d3e8a6509fd7e22d9426e664e"},"schema_version":"1.0","source":{"id":"1112.0686","kind":"arxiv","version":1}},"canonical_sha256":"54b31320a1478895e28dbc036347a589890110c9e689808881c53f2dc7243a39","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"54b31320a1478895e28dbc036347a589890110c9e689808881c53f2dc7243a39","first_computed_at":"2026-05-18T04:07:05.951624Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:07:05.951624Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"M/mO/QWC5GcC3DEf6DkH9q80Z6GVx2Z31byaFpkTDEFWxNGvuXJs9Z7uR/4Xj8x5Xb2fxlmr0pLIo6a/iQhhAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:07:05.952075Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.0686","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:afe01b657c969332445cf859b9e9701dd30e1b038cffaa04d62c913a2effe54b","sha256:960dc6f95a58675df55ea3457782a5f6656aac132e79dd8b8c76229d55a2c0e3"],"state_sha256":"e5cf16d21bf16ac12ccb676bd758d8bb0d6222a4f2f9598c81f1c6fa15981fff"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9cMjfQHDCASpGXTeCrraJIxEe2y+SKqZeCAhF+H/WpAIyu96XqJxiAWEDxwnXfjKFW+jHHQFVHMGQLRkdYm/Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T06:41:34.188993Z","bundle_sha256":"36d208058bdc0bcb3171695dc018a71d51ccfb26cb709a5b1a3f2ac8b59ccb54"}}