Iranian Mathematical Society (IMS) Bulletin of the Iranian Mathematical Society 1017-060X 38 3 2012 09 15 Optimal inequalities for the power, harmonic and logarithmic means597 606 en Yuming Chu Huzhou Teachers College Mingyu Shi Hebei University Yueping Jiang Hunan University 2009 10 16 0$, the following two optimal inequalities are presented: $H^{alpha}(a,b)L^{1-alpha}(a,b)geq M_{frac{1-4alpha}{3}}(a,b)$ for $alphain[frac{1}{4},1)$, and $ H^{alpha}(a,b)L^{1-alpha}(a,b)leq M_{frac{1-4alpha}{3}}(a,b)$ for $alphain(0,frac{3sqrt{5}-5}{40}]$. Here, $H(a,b)$, $L(a,b)$, and $M_p(a,b)$ denote the harmonic, logarithmic, and power means of order $p$ of two positive numbers $a$ and $b$, respectively.]]>