New sharp bounds for logarithmic mean and identric mean

摘要

For x , y > 0 with x ≠ y , let L = L ( x , y ) , I = I ( x , y ) , A = A ( x , y ) , G = G ( x , y ) , A r = A 1 / r ( x r , y r ) denote the logarithmic mean, identric mean, arithmetic mean, geometric mean and r-order power mean, respectively. We find the best constant p , q > 0 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.