%0 Journal Article %T Optimal inequalities for the power, harmonic and logarithmic means %J Bulletin of the Iranian Mathematical Society %I Iranian Mathematical Society (IMS) %Z 1017-060X %A Chu, Yuming %A Shi, Mingyu %A Jiang, Yueping %D 2012 %\ 09/15/2012 %V 38 %N 3 %P 597-606 %! Optimal inequalities for the power, harmonic and logarithmic means %K Power mean %K logarithmic mean %K harmonic mean %R %X For all $a,b>0$, 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. %U http://bims.iranjournals.ir/article_225_148114bdbc407891af23dda972ab8b96.pdf