We prove that αH(a, b)+(1 − α)L(a, b) > M(1−4α)/3(a, b) for α ∈ (0, 1) and all a, b > 0 with a ≠ b if and only if α ∈ [1/4, 1) and αH(a, b)+(1 − α)L(a, b) < M(1−4α)/3(a, b) if and only if , and the parameter (1 − 4α)/3 is the best possible in either case. Here, H(a, b) = 2ab/(a + b), L(a, b) = (a − b)/(log a − log b), and Mp(a, b) = ((ap + bp)/2) 1/p (p ≠ 0) and are the harmonic, logarithmic, and pth power means of a and b, respectively.