Hyperbolic functions: Difference between revisions

Revision as of 14:03, 20 December 2024

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===Inequalities===

Below are several inequalities involving hyperbolic functions, arranged from more conceptually interesting to more basic, simplified inequalities. References are preserved in the original format. Some inequalities that relate hyperbolic functions to the exponential function or provide simpler upper bounds are grouped together.

1. '''Cusa-type hyperbolic inequality:'''

<math display="block">\frac{\operatorname{sinh}(x)}{x} >< \frac{2}{3}+\frac{1}{3} \operatorname{cosh}(x), \quad x > 0.</math>

This gives a hyperbolic analogue of the classical Cusa-Huygens inequality. <ref>{{cite journal |last1=Zhu |first1=Ling |date=2010 |title=Inequalities for Hyperbolic Functions and Their Applications |journal=Journal of Inequalities and Applications |volume=2010 |page=130821 |url=https://journalofinequalitiesandapplications.springeropen.com/articles/10.1155/2010/130821}}</ref>