{"paper":{"title":"On a conjecture for trigonometric sums by S. Koumandos and S. Ruscheweyh","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"A. Swaminathan, Priyanka Sangal","submitted_at":"2018-06-19T01:26:46Z","abstract_excerpt":"S. Koumandos and S. Ruscheweyh posed the following conjecture: For $\\rho\\in(0,1]$ and $0<\\mu\\leq\\mu^{\\ast}(\\rho)$, the partial sum $s_n^{\\mu}(z)=\\displaystyle\\sum_{k=0}^n \\frac{(\\mu)_k}{k!}z^k$, $0<\\mu\\leq1$, $|z|<1$, satisfies %\n\\begin{align*} (1-z)^{\\rho}s_n^{\\mu}(z) \\prec \\left(\\frac{1+z}{1-z}\\right)^{\\rho}, \\qquad n\\in \\mathbb{N}, \\end{align*} where $\\mu^{\\ast}(\\rho)$ is the unique solution of \\begin{align*} \\int_0^{(\\rho+1)\\pi} \\sin(t-\\rho\\pi)t^{\\mu-1}dt=0. \\end{align*} This conjecture is already settled for $\\rho=\\frac{1}{2}$, $\\frac{1}{4}$, $\\frac{3}{4}$ and $\\rho=1$. In this work, we v"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.06999","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"}