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

Wu, Shan-He; Srivastava, H. M.; Debnath, Lokenath

Integral Transforms Spec. Funct. 19, No. 3, 183-193 (2008).

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.

Reviewer: V. Lokesha (Bangalore)

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

Keywords:

Jordan inequality; power series; polynomial representation; higher order derivatives; Yang Le inequality; generalizations and refinement of Jordan’s inequality

Citations:

Zbl 1132.26334; Zbl 1142.26019

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