Summary: We present various new inequalities involving the logarithmic mean , the identric mean , and the classical arithmetic and geometric means, and . In particular, we prove the following conjecture, which was published in 1986 [H. Alzer, Arch. Math. 47, 422-426 (1986; Zbl 0585.26014)]. If () denotes the power mean of order , then
with the best possible parameter .