## Sharpening Jordan’s inequality and the Yang Le inequality.(English)Zbl 1097.26012

Summary: In this work, the following inequality $\frac{\sin x}{x}\leq \frac 2\pi+ \frac{\pi-2}{\pi^3} (\pi^2-4x^2), \quad x\in (0,\pi/2]$ is established. An application of this inequality gives an improvement of the Yang Le inequality [C. J. Zhao, “Generalization and strengthening of the Yang Le inequality”, Math. Practice Theory 30, No. 4, 493–497 (2000)]: $(n-1) \sum_{k=1}^n \cos^2\lambda A_k-2\cos\lambda\pi \sum_{1\leq i<j\leq n}\cos\lambda A_i\cos\lambda A_j\leq 4\binom n2 \biggl( \lambda^3+ \frac{\lambda(1-\lambda^2)}{2} \pi\biggr)^2,$ where $$A_i>0$$ $$(i=1,2,\dots,n)$$, $$\sum_{i=1}^n A_i\leq\pi$$, $$0\leq\lambda\leq 1$$, and $$n\geq 2$$ is a natural number.

### MSC:

 26D05 Inequalities for trigonometric functions and polynomials 42A05 Trigonometric polynomials, inequalities, extremal problems

### Keywords:

upper bound; Jordan inequality; Yang Le inequality
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### References:

 [1] Mitrinovic, D. S., Analytic Inequalities (1970), Springer-Verlag · Zbl 0199.38101 [2] Debnath, L.; Zhao, C. J., New strengthened Jordan’s inequality and its applications, Appl. Math. Lett., 16, 4, 557-560 (2003) · Zbl 1041.26005 [4] Zhao, C. J., Generalization and strengthening of the Yang Le inequality, Math. Practice Theory, 30, 4, 493-497 (2000), (in Chinese) · Zbl 1493.26097 [5] Anderson, G. D.; Qiu, S.-L.; Vamanamurthy, M. K.; Vuorinen, M., Generalized elliptic integrals and modular equations, Pacific J. Math., 192, 1-37 (2000)
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