Abstract
In this paper, we present new inequalities about hyperbolic functions with much better approximation effect. It firstly provides two-sided bounds of \((\sinh (x)/x)^p\) for the case \(p \in (0,1]\), and lower bound for the case \(p \ge \frac{7}{5}\) as well. It also provides inequalities about mixed hyperbolic functions consisting of \(\tanh (x)\) and \(\sinh (x)\). Numerical examples show that the new inequalities can achieve much better approximation effect than those of prevailing methods.
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Acknowledgements
The authors would like to thank the editor and the anonymous referees for their valuable suggestions and comments which helped us to improve this paper greatly.
Funding
This research work was partially supported by Zhejiang Key Research and Development Project of China (LY19F020041, 2018C01030), the National Natural Science Foundation of China (61972120,61672009).
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Huang, W., Chen, XD., Chen, L. et al. New inequalities for hyperbolic functions based on reparameterization. RACSAM 115, 3 (2021). https://doi.org/10.1007/s13398-020-00941-0
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DOI: https://doi.org/10.1007/s13398-020-00941-0
Keywords
- Inequalities
- Inverse tangent function
- Inverse hyperbolic sine function
- Inverse hyperbolic tangent function
- Sine function