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Horst Alzer, Stamatis Koumandos

Some monotonic trigonometric sums

Keywords: trigonometric polynominals, inequalities of Fejér-Jackson and Young, monotonicity

Let

S_n(x)=∑_k=1ⁿsin(kx)/k and C_n(x)=1+∑_k=1ⁿcos(kx)/k

be the trigonometric polynomials of Fejér and Young, respectively. The aim of this paper is to determine all real parameters a and α such that for all integers n≥1 the functions

x→ S_n(x)(tan(x/2))^a and x→ C_n(x)(tan(x/2))^α

are strictly increasing on (0,π), and all parameters b and β such that for all n≥1

x→ S_n(x)(tan(x/2))^b and x→ C_n(x)(tan(x/2))^β

are strictly decreasing on (0,π). Our results complement a classical monotonicity theorem due to Askey and Steinig (1976).

Analysis, Oldenbourg Wissenschaftsverlag

Print ISSN: 0174-4747
Volume: 26, 04/2006
Pages: 429 - 449

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