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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 $f,g\\in \\mathcal{S}$"},"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":"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"},"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"},"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. 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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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1112.0686","created_at":"2026-05-18T04:07:05.951693+00:00"},{"alias_kind":"arxiv_version","alias_value":"1112.0686v1","created_at":"2026-05-18T04:07:05.951693+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.0686","created_at":"2026-05-18T04:07:05.951693+00:00"},{"alias_kind":"pith_short_12","alias_value":"KSZRGIFBI6EJ","created_at":"2026-05-18T12:26:34.985390+00:00"},{"alias_kind":"pith_short_16","alias_value":"KSZRGIFBI6EJLYUN","created_at":"2026-05-18T12:26:34.985390+00:00"},{"alias_kind":"pith_short_8","alias_value":"KSZRGIFB","created_at":"2026-05-18T12:26:34.985390+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/KSZRGIFBI6EJLYUNXQBWGR5FRG","json":"https://pith.science/pith/KSZRGIFBI6EJLYUNXQBWGR5FRG.json","graph_json":"https://pith.science/api/pith-number/KSZRGIFBI6EJLYUNXQBWGR5FRG/graph.json","events_json":"https://pith.science/api/pith-number/KSZRGIFBI6EJLYUNXQBWGR5FRG/events.json","paper":"https://pith.science/paper/KSZRGIFB"},"agent_actions":{"view_html":"https://pith.science/pith/KSZRGIFBI6EJLYUNXQBWGR5FRG","download_json":"https://pith.science/pith/KSZRGIFBI6EJLYUNXQBWGR5FRG.json","view_paper":"https://pith.science/paper/KSZRGIFB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1112.0686&json=true","fetch_graph":"https://pith.science/api/pith-number/KSZRGIFBI6EJLYUNXQBWGR5FRG/graph.json","fetch_events":"https://pith.science/api/pith-number/KSZRGIFBI6EJLYUNXQBWGR5FRG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KSZRGIFBI6EJLYUNXQBWGR5FRG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KSZRGIFBI6EJLYUNXQBWGR5FRG/action/storage_attestation","attest_author":"https://pith.science/pith/KSZRGIFBI6EJLYUNXQBWGR5FRG/action/author_attestation","sign_citation":"https://pith.science/pith/KSZRGIFBI6EJLYUNXQBWGR5FRG/action/citation_signature","submit_replication":"https://pith.science/pith/KSZRGIFBI6EJLYUNXQBWGR5FRG/action/replication_record"}},"created_at":"2026-05-18T04:07:05.951693+00:00","updated_at":"2026-05-18T04:07:05.951693+00:00"}