2023/07/10 by Horst Alzer, Man Kam Kwong, Alzer, Horst +1
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical Inequalities and Applications #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.2307.04464
openalex publication_date 2023/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present several new inequalities for trigonometric sums. Among others, we show that the inequality ∑k=1n (n-k+1)(n-k+2)ksin(kx) gt; (2)/(9) sin(x) ( 1+2cos(x) )2 holds for all n≥ 1 and x∈ (0, 2π/3). The constant factor 2/9 is sharp. This refines the classical Szegö-Schweitzer inequality which states that the sine sum is positive for all n≥ 1 and x∈ (0,2 π/3). Moreover, as an application of one of our results, we obtain a two-parameter class of absolutely monotonic functions.