Abstract
For with , let , , , , denote the logarithmic mean, identric mean, arithmetic mean, geometric mean and r-order power mean, respectively. We find the best constant such that the inequalities hold, respectively. From them some new inequalities for means are derived. Lastly, our new lower bound for the logarithmic mean is compared with several known ones, which shows that our results are superior to others. MSC: 26D07, 26E60, 05A15, 15A18.
Highlights
The logarithmic and identric means of two positive real numbers x and y with x = y are defined by x–y L = L(x, y) = ln x – ln y and I = I(x, y) = e– xx yy /(x–y), respectively
The power mean of order r of the positive real numbers x and y is defined by xr + yr /r
The logarithmic mean and the identric mean have been the subject of intensive research
Summary
We need the following lemma, which tells us an inequality for bivariate homogeneous means can be equivalently changed into the form of hyperbolic functions. The inequality sinh t > (cosh pt) /( p ) t holds for all t if and only if p and the function p Proof of Theorem 1 In order to prove Theorem , we first give the following lemmas. Lemma For t > , let the function U : ( , ∞) → ( , ∞) be defined by
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