Details

\((i) \qquad \lambda \, \sum_{k=1}^n (-1)^k \sin(kx) \leq \sum_{k=1}^n \frac{\sin(kx)}{k} \)

is valid for all \(n\geq 1\) and \(x\in [0,\pi]\) if and only if \(\lambda \in [0,2]\). This extends the classical Fejér-Jackson inequality which states that \((i)\) holds for \(\lambda=0\). An application of \((i)\) reveals if \(a>0\) and \(b\) are real numbers, then

\((ii) \qquad \frac{41}{96}+ \sum_{k=1}^n \frac{\cos(kx)}{k+1} \geq a \bigl( \cos(x)+b\bigr)^2 \)

holds for all \(n\geq 2\) and \(x\in[0,\pi]\) if and only if \(a\leq 2/75\) and \(b=3/8\). This refines a result of Koumandos (2001) who proved that the expression on the left-hand side of \((ii)\) is nonnegative for all \(n\geq 2\) and \(x\in[0,\pi]\). The cosine polynomial in \((ii)\) was first studied by Rogosinski and Szegö in 1928.

References

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