@article {
author = {Chu, Yuming and Shi, Mingyu and Jiang, Yueping},
title = {Optimal inequalities for the power, harmonic and logarithmic means},
journal = {Bulletin of the Iranian Mathematical Society},
volume = {38},
number = {3},
pages = {597-606},
year = {2012},
publisher = {Iranian Mathematical Society (IMS)},
issn = {1017-060X},
eissn = {1735-8515},
doi = {},
abstract = {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.},
keywords = {Power mean,logarithmic mean,harmonic mean},
url = {http://bims.iranjournals.ir/article_225.html},
eprint = {http://bims.iranjournals.ir/article_225_148114bdbc407891af23dda972ab8b96.pdf}
}