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We prove that a function $f(z)=\\sum_{n=0}^\\infty a_nz^n$ with $a_n\\downarrow 0$ is in $\\bk^1_{\\log^\\alpha}$ iff $\\sum_{n=0}^\\infty \\log^\\alpha(n+2)/(n+1)<\\infty$ and apply this to obtain a criterion for membership of the Libera transform of a function with positive coefficients in $\\bk"},"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":"1104.4629","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2011-04-24T13:24:04Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"9d1d4c882887f951bb8e389b3204207170bef07cd50764452af6c5a24e6152c1","abstract_canon_sha256":"ee54e89c366afa06ca0da7f520e98198a3a4a1faf7ef93266f3cb69af8ae24d1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:23:33.733637Z","signature_b64":"/1d0zY+nw7wp1lIvUolvqS2tZBivE48DYnHloXY1+iCiD2jNSO2WLyP+jvztwv5dP8JURJExHy/LnujR1utaAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5b09d652df6989e026e6129f5a092b40413848cbfa71926a0240283b3eda5f4f","last_reissued_at":"2026-05-18T04:23:33.733001Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:23:33.733001Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Logarithmic Bloch space and its predual","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CV","authors_text":"Miroslav Pavlovi\\'c","submitted_at":"2011-04-24T13:24:04Z","abstract_excerpt":"We consider the space $\\bk^1_{\\log^\\alpha}$, of analytic functions on the unit disk $\\D,$ defined by the requirement $\\int_\\D|f'(z)|\\phi(|z|)\\,dA(z)<\\infty,$ where $\\phi(r)=\\log^\\alpha(1/(1-r))$ and show that it is a predual of the \"$\\log^\\alpha$-Bloch\" space and the dual of the corresponding little Bloch space. We prove that a function $f(z)=\\sum_{n=0}^\\infty a_nz^n$ with $a_n\\downarrow 0$ is in $\\bk^1_{\\log^\\alpha}$ iff $\\sum_{n=0}^\\infty \\log^\\alpha(n+2)/(n+1)<\\infty$ and apply this to obtain a criterion for membership of the Libera transform of a function with positive coefficients in $\\bk"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.4629","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":"1104.4629","created_at":"2026-05-18T04:23:33.733105+00:00"},{"alias_kind":"arxiv_version","alias_value":"1104.4629v1","created_at":"2026-05-18T04:23:33.733105+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.4629","created_at":"2026-05-18T04:23:33.733105+00:00"},{"alias_kind":"pith_short_12","alias_value":"LME5MUW7NGE6","created_at":"2026-05-18T12:26:34.985390+00:00"},{"alias_kind":"pith_short_16","alias_value":"LME5MUW7NGE6AJXG","created_at":"2026-05-18T12:26:34.985390+00:00"},{"alias_kind":"pith_short_8","alias_value":"LME5MUW7","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/LME5MUW7NGE6AJXGCKPVUCJLIB","json":"https://pith.science/pith/LME5MUW7NGE6AJXGCKPVUCJLIB.json","graph_json":"https://pith.science/api/pith-number/LME5MUW7NGE6AJXGCKPVUCJLIB/graph.json","events_json":"https://pith.science/api/pith-number/LME5MUW7NGE6AJXGCKPVUCJLIB/events.json","paper":"https://pith.science/paper/LME5MUW7"},"agent_actions":{"view_html":"https://pith.science/pith/LME5MUW7NGE6AJXGCKPVUCJLIB","download_json":"https://pith.science/pith/LME5MUW7NGE6AJXGCKPVUCJLIB.json","view_paper":"https://pith.science/paper/LME5MUW7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1104.4629&json=true","fetch_graph":"https://pith.science/api/pith-number/LME5MUW7NGE6AJXGCKPVUCJLIB/graph.json","fetch_events":"https://pith.science/api/pith-number/LME5MUW7NGE6AJXGCKPVUCJLIB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LME5MUW7NGE6AJXGCKPVUCJLIB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LME5MUW7NGE6AJXGCKPVUCJLIB/action/storage_attestation","attest_author":"https://pith.science/pith/LME5MUW7NGE6AJXGCKPVUCJLIB/action/author_attestation","sign_citation":"https://pith.science/pith/LME5MUW7NGE6AJXGCKPVUCJLIB/action/citation_signature","submit_replication":"https://pith.science/pith/LME5MUW7NGE6AJXGCKPVUCJLIB/action/replication_record"}},"created_at":"2026-05-18T04:23:33.733105+00:00","updated_at":"2026-05-18T04:23:33.733105+00:00"}