Inequalities for alternating trigonometric sums. (English) Zbl 1272.26008
Summary: J. B. Kelly [Inequalities 2, Proc. 2nd Sympos. Inequalities, U.S. Air Force Acad., Colorado 1967, 193–212 (1970; Zbl 0223.52014)] proved that
\[ 0\leq\sum\limits_{k=1}^n (-1)^{k+1}(n-k+1)|\sin(kx)|\quad(n\in\mathbf N;x\in\mathbf R). \] We generalize and complement this inequality. Moreover, we present sharp upper and lower bounds for the related sums \[ \begin{aligned} &\sum\limits_{k=1}^n(-1)^{k+1}(n-k+1)|\cos(kx)|\quad \text{ and }\\ &\sum\limits_{k=1}^n(-1)^{k+1}(n-k+1)\bigl(|\sin(kx)| +|\cos(kx)|\bigr).\end{aligned} \]
\[ 0\leq\sum\limits_{k=1}^n (-1)^{k+1}(n-k+1)|\sin(kx)|\quad(n\in\mathbf N;x\in\mathbf R). \] We generalize and complement this inequality. Moreover, we present sharp upper and lower bounds for the related sums \[ \begin{aligned} &\sum\limits_{k=1}^n(-1)^{k+1}(n-k+1)|\cos(kx)|\quad \text{ and }\\ &\sum\limits_{k=1}^n(-1)^{k+1}(n-k+1)\bigl(|\sin(kx)| +|\cos(kx)|\bigr).\end{aligned} \]
MSC:
| 26D05 | Inequalities for trigonometric functions and polynomials |
| 26D15 | Inequalities for sums, series and integrals |
| 54E35 | Metric spaces, metrizability |
Citations:
Zbl 0223.52014References:
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