{"paper":{"title":"Integral mean estimates for the polar derivative of a polynomial","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"N. A. Rather, Suhail Gulzar","submitted_at":"2013-02-28T03:22:19Z","abstract_excerpt":"Let $ P(z) $ be a polynomial of degree $ n $ having all zeros in $|z|\\leq k$ where $k\\leq 1,$ then it was proved by Dewan \\textit{et al} that for every real or complex number $\\alpha$ with $|\\alpha|\\geq k$ and each $r\\geq 0$\n  $$ n(|\\alpha|-k)\\left\\{\\int\\limits_{0}^{2\\pi}\\left|P\\left(e^{i\\theta}\\right)\\right|^r d\\theta\\right\\}^{\\frac{1}{r}}\\leq\\left\\{\\int\\limits_{0}^{2\\pi}\\left|1+ke^{i\\theta}\\right|^r d\\theta\\right\\}^{\\frac{1}{r}}\\underset{|z|=1}{Max}|D_\\alpha P(z)|. $$\n  \\indent In this paper, we shall present a refinement and generalization of above result and also extend it to the class of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.7066","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"}