{"paper":{"title":"Taylor Domination, Tur\\'an lemma, and Poincar\\'e-Perron Sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Dmitry Batenkov, Yosef Yomdin","submitted_at":"2013-01-25T12:55:05Z","abstract_excerpt":"We consider \"Taylor domination\" property for an analytic function $f(z)=\\sum_{k=0}^{\\infty}a_{k}z^{k},$ in the complex disk $D_R$, which is an inequality of the form \\[ |a_{k}|R^{k}\\leq C\\ \\max_{i=0,\\dots,N}\\ |a_{i}|R^{i}, \\ k \\geq N+1. \\] This property is closely related to the classical notion of \"valency\" of $f$ in $D_R$. For $f$ - rational function we show that Taylor domination is essentially equivalent to a well-known and widely used Tur\\'an's inequality on the sums of powers.\n  Next we consider linear recurrence relations of the Poincar\\'e type \\[ a_{k}=\\sum_{j=1}^{d}[c_{j}+\\psi_{j}(k)]"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.6033","kind":"arxiv","version":2},"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"}