For all a, b > 0, the following two optimal inequalities are presented: Hα(a, b)L1-α(a, b) ≥ M1-4α/3(a, b) for α ∈ [1/4, 1), and Hα(a, b)L1-α(a, b) ≤ M1-4α/3(a, b) for α ∈ (0, 3√5-5/40]. Here, H(a, b), L(a, b), and Mp(a, b) denote the harmonic, logarithmic, and power means of order p of two positive numbers a and b, respectively.