We present various new inequalities involving the logarithmic mean L (x, y) = (x-y)/(log x-log y), the identric mean I (x, y) = (1/e) (x~x/y~y)~(1 (x-y)), and the classical arithmetic and geometric means, A (x, y) = (x+y)/2 and G (x, y) = (xy)~(1/2). In particular, we prove the following conjecture, which was published in 1986 in this journal. If M_r (x, y) = (x~r/2 + y~r/2)~(1/r) (r ≠ 0) denotes the power mean of order r, then M_c (x, y) < 1/2 L (x, y) + I (x, y) (x, y > 0, x ≠ y), with the best possible parameter c = (log 2)/(1 + log 2).
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