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Zbl 1226.26019
Qiu, Ye-Fang; Wang, Miao-Kun; Chu, Yu-Ming; Wang, Gen-Di
Two sharp inequalities for Lehmer mean, identric mean and logarithmic mean.
(English)
[J] J. Math. Inequal. 5, No. 3, 301-306 (2011). ISSN 1846-579X

Summary: For $r\in\Bbb R$, the Lehmer mean of two positive numbers $a$ and $b$ is defined by $$L_r(a,b)= \frac{a^{r+1}+b^{r+1}}{a^r+b^r}.$$ In this paper, we establish two sharp inequalities as follows: $I(a,b)> L _{-\frac16}(a,b)$ and $L(a,b)>L_{-\frac13}(a,b)$ for all $a,b>0$ with $a\ne b$. Here, $I(a,b)= \frac1e (\frac{b^b}{a^a})^{\frac{1}{b-a}}$ and $L(a,b)= \frac{b-a}{\log b-\log a}$ denote the identric mean and the logarithmic mean of two positive numbers $a$ and $b$ with $a\ne b$, respectively.
MSC 2000:
*26E60 Means

Keywords: Lehmer mean; identric mean; logarithmic mean

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