A sharp inequality involving hyperbolic and inverse hyperbolic functions

Abstract

We prove that the inequality $$\cosh \left(\mathrm{arcosh}(2\cosh u)\cdot \tanh u\right)<\exp \left(u\cdot \tanh u\right)$$ holds for all $u>0$. We check with the computation program Mathematica that the ratio between the left-hand and the right-hand side is greater than 0,97 for all $u\ge 0$, so this is a quite sharp inequality. It is also equivalent to any of the two inequalities: $$\cosh \left(\sqrt{1-\frac{1}{t^2}}\cdot \mathrm{arcosh}2t\right)<\exp \left(\sqrt{1-\frac{1}{t^2}}\cdot \mathrm{arcosh}t\right)$$ for all $t>1$, and $$\cosh \left(c\cdot \mathrm{arcosh}\frac{2}{\sqrt{1-c^2}}\right)<\exp \left(c\cdot \mathrm{arcosh}\frac{1}{\sqrt{1-c^2}}\right)$$ for all $c\in (0,1)$.

4 pages