caa a22 4500 463177967 CHVBK 20180406164830.0 cr unu---uuuuu 170326e20071201xx s 000 0 eng 10.1007/s11139-007-9036-6 doi (NATIONALLICENCE)springer-10.1007/s11139-007-9036-6 Extensions of Vietoris's inequalities I [Elektronische Daten] [Gavin Brown, Feng Dai, Kunyang Wang] Let β 0=0.308443 denote the Littlewood-Salem-Izumi number, i.e., the unique solution of the equation $$\int_{0}^{\frac{3}{2}\pi}\frac {\cos t}{t^{\beta}}\,dt=0.$$ In this paper it is proved that if a 0≥a 1≥⋅⋅⋅≥a n >0 and $a_{2k}\leq(1-\frac {\beta _{0}}{k})a_{2k-1}$ , k≥1 then for all θ∈(0,π) $$\sum_{k=0}^{n}a_{k}\cos k\theta >0,$$ and furthermore, the number β 0 is best possible in the sense that for any β∈(0,β 0) $$\lim_{n\to\infty}\min_{\theta \in(0,\pi)}\sum_{k=0}^{n}c_{k}(\beta )\cos k\theta =-\infty,$$ where the coefficients c k (β) are defined as $$c_{0}(\beta )=c_{1}(\beta )=1,\qquad c_{2k}(\beta )=c_{2k+1}(\beta )=\biggl(1-\frac{\beta}{k}\biggr)c_{2k-1}(\beta ),\quad k\ge1.$$ Results for the sine sums are obtained as well. These results generalize and sharpen the well known trigonometric inequalities of Vietoris. Springer Science+Business Media, LLC, 2007 Vietoris's inequalities nationallicence Positive trigonometric sums nationallicence Brown Gavin Office of the Vice-Chancellor and Principal, A14 - Main Quadrangle, The University of Sydney, 2006, Sydney, NSW, Australia aut Dai Feng Department of Mathematical and Statistical Sciences, CAB 632, University of Alberta, T6G 2G1, Edmonton, AB, Canada aut Wang Kunyang Department of Mathematics, Beijing Normal University, 100875, Beijing, China aut The Ramanujan Journal Springer US; http://www.springer-ny.com 14/3(2007-12-01), 471-507 1382-4090 14:3<471 2007 14 11139 https://doi.org/10.1007/s11139-007-9036-6 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11139-007-9036-6 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Brown Gavin Office of the Vice-Chancellor and Principal, A14 - Main Quadrangle, The University of Sydney, 2006, Sydney, NSW, Australia aut NATIONALLICENCE 700 1- Dai Feng Department of Mathematical and Statistical Sciences, CAB 632, University of Alberta, T6G 2G1, Edmonton, AB, Canada aut NATIONALLICENCE 700 1- Wang Kunyang Department of Mathematics, Beijing Normal University, 100875, Beijing, China aut NATIONALLICENCE 773 0- The Ramanujan Journal Springer US; http://www.springer-ny.com 14/3(2007-12-01), 471-507 1382-4090 14:3<471 2007 14 11139 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer