A sharp bound for a sine polynomial
Volume 96 / 2003
Colloquium Mathematicum 96 (2003), 83-91
MSC: Primary 26D05, 42A05; Secondary 26D15.
DOI: 10.4064/cm96-1-8
Abstract
We prove that $$ \left|\sum_{k=1}^{n}\frac{\sin((2k-1)x)}{k}\right| < {\rm Si}(\pi)=1.8519\dots $$ for all integers $n\geq 1$ and real numbers $x$. The upper bound ${\rm Si}(\pi)$ is best possible. This result refines inequalities due to Fejér (1910) and Lenz (1951).
