## 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.

### MSC:

 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

### Citations:

Zbl 1132.26334; Zbl 1142.26019
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### References:

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