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Then given arbitrary small $\\delta\\in (0,1)$ and $\\varepsilon >0$ the relation of these functions admits the estimate $|\\psi_1(z)/\\psi_2(z) -1| \\leqslant \\varepsilon R^{-\\mu(1-\\delta)}$ for all $|z|\\leqslant R^{1-\\delta}$, provided "},"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":"1601.04696","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2016-01-16T11:29:17Z","cross_cats_sorted":[],"title_canon_sha256":"858794b91d4afaef12ca5bda2b0b140457233952a179f0dc2f1971f239638f05","abstract_canon_sha256":"61445a85c1de9f0b62ff36086f873a6c5f09bcbc1c469fb64f51e1518b4728ff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:31.984611Z","signature_b64":"95gznLr5vTZvUmjG3FajdoE5Hy9bEwRRbFTtXRT4h9r2fGc+ZsJ3CaJXBzUqM7O+M2KEEXwDCLG5eFIH8Wx1DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62437e15e68cdbc76be221b94e5b8941e4dabd029b5497331efd945383ea97f3","last_reissued_at":"2026-05-18T01:22:31.984166Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:31.984166Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The estimation of the ratio of two entire functions with the same zeros in the ball","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"A.A.Shkalikov, V.L.Geynts","submitted_at":"2016-01-16T11:29:17Z","abstract_excerpt":"The paper studies entire functions of finite order of growth for which a representation of the form $\\psi(z) = 1+ O(|z|^{-\\mu}), \\mu >0,$ as $z\\to \\infty$, is valid on a fixed ray of the complex plane. 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