TY - JOUR ID - 225 TI - Optimal inequalities for the power, harmonic and logarithmic means JO - Bulletin of the Iranian Mathematical Society JA - BIMS LA - en SN - 1017-060X AU - Chu, Yuming AU - Shi, Mingyu AU - Jiang, Yueping AD - Huzhou Teachers College AD - Hebei University AD - Hunan University Y1 - 2012 PY - 2012 VL - 38 IS - 3 SP - 597 EP - 606 KW - Power mean KW - logarithmic mean KW - harmonic mean DO - N2 - 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. UR - http://bims.iranjournals.ir/article_225.html L1 - http://bims.iranjournals.ir/article_225_148114bdbc407891af23dda972ab8b96.pdf ER -