×
Li, Jian-Lin

An identity related to Jordan’s inequality. (English) Zbl 1143.26010

Int. J. Math. Math. Sci. 2006, No. 18, 76782, 6 p. (2006).
The paper establishes an identity that expresses \(\frac {\sin(x)} x\) as an infinite series of \((\pi^2 -4x^2)\). The partial sums of the series are then used to estimate \(\frac {\sin(x)} x\), which generalize the left hand side of Jordan’s inequality \[ \frac 2 \pi \leq \frac {\sin(x)} x < 1, \qquad x \in (0, \pi / 2]. \] The estimates are stated stronger than existing results. The paper also uses the Taylor expansion of \(\frac x {\sin(x)}\) to generalize the right hand side of Jordan’s inequality. The reciprocal of a partial sum of this series is used as an estimate on the right hand side.
Reviewer: Tieling Chen (Aiken)

Cited in 22 Documents

MSC:

26D15 Inequalities for sums, series and integrals

Keywords:

Taylor expansion
PDF BibTeX XML Cite
\textit{J.-L. Li}, Int. J. Math. Math. Sci. 2006, No. 18, 76782, 6 p. (2006; Zbl 1143.26010)
Full Text: DOI EuDML

References:

[1] Anderson, G. D.; Qiu, S.-L.; Vamanamurthy, M. K.; Vuorinen, M., Generalized elliptic integrals and modular equations, Pacific Journal of Mathematics, 192, 1, 1-37 (2000)
[2] Debnath, L.; Zhao, C.-J., New strengthened Jordan’s inequality and its applications, Applied Mathematics Letters, 16, 4, 557-560 (2003) · Zbl 1041.26005
[3] Erdélyi, A., Higher Transcental Functions, Vol. I (1953), New York: McGraw-Hill, New York
[4] Mercer, A. McD.; Abel, U.; Caccia, D., A sharpening of Jordan’s inequality, The American Mathematical Monthly, 93, 568-569 (1986)
[5] Mitrinović, D. S.; Vasić, P. M., Analytic Inequalities, xii+400 (1970), New York: Springer, New York
[6] Özban, A. Y., A new refined form of Jordan’s inequality and its applications, Applied Mathematics Letters, 19, 2, 155-160 (2006) · Zbl 1109.26011
[7] Wu, S.; Debnath, L., A new generalized and sharp version of Jordan’s inequality and its application to the improvement of Yang Le inequality, Applied Mathematics Letters, 19, 12, 1378-1384 (2006) · Zbl 1132.26334
[8] Wu, S.; Debnath, L., A new generalized and sharp version of Jordan’s inequality and its application to the improvement of Yang Le inequality II · Zbl 1132.26334
[9] Zhu, L., Sharpening Jordan’s inequality and the Yang Le inequality, Applied Mathematics Letters, 19, 3, 240-243 (2006) · Zbl 1097.26012
[10] Zhu, L., Sharpening Jordan’s inequality and Yang Le inequality. II, Applied Mathematics Letters, 19, 9, 990-994 (2006) · Zbl 1122.26014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.