On certain inequalities for hyperbolic and trigonometric functions (Q2855180)

The author obtains several inequalities betwen the trigonometric and hyperbolic functions amd their inverses. Typical results are: if \(-1<x<1\) then \(\sin x \arcsin x\geq x^2\) and if \(x\in \mathbb R\) then \(\sinh x \mathrm{ arcsinh }x\geq x^2\). These results are based on a general inequality between a function and its inverse; if \(f\) is a bijection on an interval in \(]0, \infty[\), with \(f(x)/x\) strictly increasing then \(f(x)\geq y \Longrightarrow f(x) f^{-1}(y) \geq xy\).