Some refined families of Jordan-type inequalities and their applications. (English) Zbl 1147.26017

The authors reconsider classical Jordan inequalities and establish a new sharpened and generalized version of Jordan-type inequality by means of polynomial representation as follows. Let \(0<x\leq\theta\leq\pi\) then for any natural number \(n\), the following inequality holds true: \[ \begin{split}\sum_{k=0}^n\frac{f(\theta^2)}{k!}(x^2-\theta)^K\leq\frac{\sin x}{x}\\ \leq \sum_{k=0}^{n-1} \frac{f^k(\theta)^2(x^2-\theta^2)^K}{k!}+ \left(\frac{-1}{\theta^2}\right) \left(1-\sum_{k=0}^{n-1} \frac{(-\theta^2)^kf^k(\theta^2)}{k!}\right) (x^2-\theta^2)^n\end{split} \] where \(f(x)=\frac{\sin\sqrt{x}}{\sqrt{x}}\). Furthermore, the equalities hold true if and only if \(x=\theta\).
Finally, an application of the results presented in this paper towards the improvement of the Yang Le inequality [S. Wu and L. Debnath, Appl. Math. Lett. 19, No. 12, 1378–1384 (2006; Zbl 1132.26334) and the first two authors, Appl. Math. Comput. 197, No. 2, 914–923 (2008; Zbl 1142.26019)] is considered. The authors mention at the end several other interesting consequences and applications of the results in this paper.
This paper contains still more informations related to Jordan-type inequalities and their applications.


26D15 Inequalities for sums, series and integrals
26D05 Inequalities for trigonometric functions and polynomials
26D07 Inequalities involving other types of functions
33B10 Exponential and trigonometric functions
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[1] Anderson G. D., Conformal Invariants, Inequalities, and Quasiconformal Maps (1997) · Zbl 0885.30012
[2] DOI: 10.2140/pjm.2000.192.1 · Zbl 0951.33012
[3] DOI: 10.1016/S0893-9659(03)00036-3 · Zbl 1041.26005
[4] DOI: 10.2307/2323042
[5] Mitrinović D. S., Analytic Inequalities (1970)
[6] DOI: 10.1016/j.aml.2005.05.003 · Zbl 1109.26011
[7] Wu S.-H., Octogon Math. Mag. 12 pp 267– (2004)
[8] Wu S.-H., Taiwanese J. Math. 12 (2) (2008)
[9] DOI: 10.1016/j.aml.2006.02.005 · Zbl 1132.26334
[10] DOI: 10.1016/j.aml.2006.05.022 · Zbl 1162.26310
[11] DOI: 10.1080/10652460701284164 · Zbl 1128.26017
[12] DOI: 10.1016/j.amc.2007.03.020 · Zbl 1193.26025
[13] Wu S.-H., Appl. Math. Comput. (2008)
[14] Yang L., Distribution of Values and New Research (1982)
[15] Yuefeng F., Math. Mag. 69 pp 126– (1996)
[16] Zhao C.-J., J. Inequal. Pure Appl. Math. 3 (4) pp 1– (2002)
[17] DOI: 10.1016/j.aml.2005.06.004 · Zbl 1097.26012
[18] Zhu L., Math. Inequal. Appl. 9 pp 103– (2006)
[19] DOI: 10.1016/j.aml.2005.11.011 · Zbl 1122.26014
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