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Niu, Da-Wei; Huo, Zhen-Hong; Cao, Jian; Qi, Feng

A general refinement of Jordan’s inequality and a refinement of L. Yang’s inequality. (English) Zbl 1141.26003

Integral Transforms Spec. Funct. 19, No. 3, 157-164 (2008).
Summary: In the article, a general double refinement of Jordan’s inequality,
\[ \sum^n_{k=1}\alpha_k(\pi^2-4x^2)^k\leq \frac{\sin x}{x}-\frac 2\pi\leq\sum^n_{k=1}\beta_k(\phi^2-4x^2)^k \]
for \(n\in\mathbb N\), is established, where the coefficients \(\alpha_k\) und \(\beta_k\) defined by recursing formulas (9) and (10) are the best possible. As an application, L. Yang’s inequality is refined.

Cited in 16 Documents

MSC:

26D05 Inequalities for trigonometric functions and polynomials
26D15 Inequalities for sums, series and integrals

Keywords:

Jordan’s inequality; L. Yang’s inequality; L’Hôspital’s rule; refinement; application
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\textit{D.-W. Niu} et al., Integral Transforms Spec. Funct. 19, No. 3, 157--164 (2008; Zbl 1141.26003)
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