Technical Details of the Proof of the Sine Inequality [1.2ex] \normalsize ∑_k=1^n-1( (n)/(k) - (k)/(n) ) ^βsin(kx) ≥ 0

2017/02/11 by Man Kam Kwong, Kwong, Man Kam
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.1702.03387

openalex publication_date 2017/02/11 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

In a recent study, H. Alzer and the author showed that the sine polynomial ∑_k=1^n-1 ( (n)/(k) - (k)/(n) ) ^β sin(kx) gt; 0 is nonnegative for x∈[0,π] , n≥ 2, β≥ β₁ := (log(2))/(log(16/5)) . This result, among others, will be presented in a forthcoming article. The proof relies on quite a number of technical Lemmas and inequalities. We have decided to delegate all the tedious details of the proofs of these Lemmas in a separate article, namely, the current one. Some of the proofs require brute-force numerical computation, performed with the help of the computer software MAPLE. A few of the Lemmas included here are of independent interest.

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