Inequalities between power means and convex combinations of the harmonic and logarithmic means (Q410979)

Summary: We prove that \(\alpha H(a, b) + (1 - \alpha)L(a, b) > M_{(1-4\alpha)/3} (a, b)\) for \(\alpha \in (0, 1)\) and all \(a, b > 0\) with \(a \neq b\) if and only if \(\alpha \in [1/4, 1)\) and \(\alpha H(a, b) + (1 - \alpha)L(a, b) < M_{(1-4\alpha)/3} (a, b)\) if and only if \(\alpha \in (0,3 \sqrt{345}/80 - 11/16)\), and the parameter \((1 - 4\alpha)/3\) is the best possible in either case. Here, \(H(a, b) = 2ab/(a + b), L(a, b) = (a - b)/(\log a - \log b)\), and \(M_p(a, b) = ((a^p + b^p)/2)^{1/p}(p \neq 0)\) and \(M_0 (a, b) = \sqrt{ab}\) are the harmonic, logarithmic, and \(p\)th power means of \(a\) and \(b\), respectively.