2018/09/04 by Roman Drnovšek, Drnovšek, Roman
Mathematics · #Functional Equations Stability Results #Mathematical Inequalities and Applications #Mathematics and Applications #math.CA #msc:26D07
paper · pdf · doi:10.48550/arxiv.1809.08974
4 pages
arxiv created 2018/09/04 · arxiv updated 2018/09/25
We prove that the inequality \cosh ( arcosh(2 \cosh u) ⋅ \tanh u ) < exp ( u ⋅ \tanh u ) 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 ≥ 0, so this is a quite sharp inequality. It is also equivalent to any of the two inequalities: \cosh ( √(1 - (1)/(t2)) ⋅ arcosh 2t ) < exp ( √(1 - (1)/(t2)) ⋅ arcosh t ) for all t > 1, and \cosh ( c ⋅ arcosh(2)/(√(1-c2)) ) < exp ( c ⋅ arcosh(1)/(√(1-c2)) ) for all c ∈ (0,1).