On a trigonometric inequality of Askey and Steinig

Article type: Research Article

Authors: Alzer, Horst^{a; *} | Kwong, Man Kam^{b; **}

Affiliations: [a] Morsbacher Straße 10, 51545 Waldbröl, Germany. E-mail: h.alzer@gmx.de | [b] Department of Mathematics, The Hong Kong Polytechnic University, Hunghom, Hong Kong. E-mail: mankwong@polyu.edu.hk

Correspondence: [*] Corresponding author. E-mail: h.alzer@gmx.de.

Note: [**] The research of this author is supported by the Hong Kong Government GRF Grant PolyU 5003/12P and the Hong Kong Polytechnic University Grants G-UC22 and G-UA10.

Abstract: In 1974, Askey and Steinig showed that for n⩾0 and x∈(0,2π), Sn(x)=∑k=0nsin((k+1/4)x)k+1>0andCn(x)=∑k=0ncos((k+1/4)x)k+1>0. We prove that (0.1)Sn(x)+Cn(x)⩾12 and that the alternating sums Sn∗(x)=∑k=0n(−1)ksin((k+1/4)x)k+1andCn∗(x)=∑k=0n(−1)kcos((k+1/4)x)k+1 satisfy (0.2)Sn∗(x)+Cn∗(x)⩾1200(13−85)300+2085=0.41601…. Both inequalities hold for all n⩾0 and x∈[0,2π]. The constant lower bounds given in (0.1) and (0.2) are best possible. The asymptotic behaviour of both sums is also investigated.

Keywords: Trigonometric sums, inequalities, asymptotics

DOI: 10.3233/ASY-171447

Journal: Asymptotic Analysis, vol. 106, no. 3-4, pp. 233-249, 2018