SHARP POWER MEAN BOUNDS FOR THE TANGENT AND HYPERBOLIC SINE MEANS

Abstract

In the article, we prove that the double inequalities Mα1 (a,b) < Mtan(a,b) < Mβ1 (a,b), Mα2 (a,b) < Msinh(a,b) < Mβ2 (a,b) hold for all a,b > 0 with a ≠ b if and only if α1 ≤ 1/3, β1 ≥ log2/log(2tan1) ≈ 0.61007, α2 ≤ 2/3 and β2 ≥ log2/log(2sinh1) ≈ 0.81109, where Mp, Mtan and Msinh are the pth power mean, tangent mean and hyperbolic sine mean, respectively.