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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 theleft-hand and the right-hand side is greater than 0,97 for all $u \ge 0$, sothis is a quite sharp inequality. It is also equivalent to any of the twoinequalities: $$ \cosh \left( \sqrt{1 - \frac{1}{t^2}} \cdot\mathrm{arcosh}\,{2t} \right) < \exp \left( \sqrt{1 - \frac{1}{t^2}} \cdot\mathrm{arcosh}\,{t} \right) $$ 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 $c \in (0,1)$.

A sharp inequality involving hyperbolic and inverse hyperbolic functions