{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:MBT3VWBAV4I3L5XJ66RHUNXH7N","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":"8cd3e515d0708c8df710c8c78bce1cf0a7e0bbaf69e6fa128d99309a94165f77","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-02-26T13:46:41Z","title_canon_sha256":"7a658046a3ed062ec019b5c8d586a5d3c17817b3d2c76b9e629711bd4c9a2d88"},"schema_version":"1.0","source":{"id":"1402.6539","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.6539","created_at":"2026-05-18T02:57:44Z"},{"alias_kind":"arxiv_version","alias_value":"1402.6539v1","created_at":"2026-05-18T02:57:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.6539","created_at":"2026-05-18T02:57:44Z"},{"alias_kind":"pith_short_12","alias_value":"MBT3VWBAV4I3","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"MBT3VWBAV4I3L5XJ","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"MBT3VWBA","created_at":"2026-05-18T12:28:38Z"}],"graph_snapshots":[{"event_id":"sha256:b860eb58de3004a16218a46409a7b14426939896add144f8eb061c1100b6ae9d","target":"graph","created_at":"2026-05-18T02:57:44Z","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":"A recent conjecture by I. Ra\\c{s}a asserts that the sum of the squared Bernstein basis polynomials is a convex function in $[0,1]$. This conjecture turns out to be equivalent to a certain upper pointwise estimate of the ratio $P_n^{\\prime}(x)/P_n(x)$ for $x\\geq 1$, where $P_n$ is the $n$-th Legendre polynomial. Here, we prove both upper and lower pointwise estimates for the ratios $\\big(P_n^{(\\lambda)}(x)\\big)^{\\prime}/P_n^{(\\lambda)}(x)$, $~x\\geq 1$, where $P_n^{(\\lambda)}$ is the $n$-th ultraspherical polynomial. In particular, we validate Ra\\c{s}a's conjecture.","authors_text":"Geno Nikolov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-02-26T13:46:41Z","title":"Inequalities for ultraspherical polynomials. Proof of a conjecture of I. Ra\\c{s}a"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.6539","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":"sha256:97c053cabbf2c007f4cff6705a755a03288b6f6db1c91f7fc996552f643e0d24","target":"record","created_at":"2026-05-18T02:57:44Z","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":"8cd3e515d0708c8df710c8c78bce1cf0a7e0bbaf69e6fa128d99309a94165f77","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2014-02-26T13:46:41Z","title_canon_sha256":"7a658046a3ed062ec019b5c8d586a5d3c17817b3d2c76b9e629711bd4c9a2d88"},"schema_version":"1.0","source":{"id":"1402.6539","kind":"arxiv","version":1}},"canonical_sha256":"6067bad820af11b5f6e9f7a27a36e7fb69a7665023925927e229df4eb7a6c259","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6067bad820af11b5f6e9f7a27a36e7fb69a7665023925927e229df4eb7a6c259","first_computed_at":"2026-05-18T02:57:44.325818Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:57:44.325818Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nEYHueaKpCtCDvWo20nYAEArKK6TRxaOhPQEAygjxAeQ4XtZJJT2ha3LRMY3h/b+fweGxYeCbCa8APgbtioGCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:57:44.326300Z","signed_message":"canonical_sha256_bytes"},"source_id":"1402.6539","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:97c053cabbf2c007f4cff6705a755a03288b6f6db1c91f7fc996552f643e0d24","sha256:b860eb58de3004a16218a46409a7b14426939896add144f8eb061c1100b6ae9d"],"state_sha256":"81e1b7f03f6decba5fd5cc93d47eb6461bfa029e4d6779b6563da5565d3897fd"}