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Zbl 1187.26018
Wang, Miao-Kun; Chu, Yu-Ming; Qiu, Ye-Fang
Some comparison inequalities for generalized Muirhead and identric means.
(English)
[J] J. Inequal. Appl. 2010, Article ID 295620, 10 p. (2010). ISSN 1029-242X/e

Summary: For $x,y>0$, $a,b\in\Bbb R$, with $a+b\ne0$, the generalized Muirhead mean $M(a,b;x,y)$ with parameters $a$ and $b$ and the identric mean $I(x,y)$ are defined by $M(a,b;x,y)= ((x^ay^b+x^by^a)/2)^{1/(a+b)}$ and $I(x,y)= (1/e)(y^y/x^x)^{1/(y-x)}$, $x\ne y$, $I(x,y)=x$, $x=y$, respectively. In this paper, the following results are established: {\parindent=7mm \item{(1)} $M(a,b;x,y)> I(x,y)$ for all $x,y>0$ with $x\ne y$ and $(a,b)\in \{(a,b)\in\Bbb R^2: a+b>0$, $ab\le 0$, $2(a-b)^2 -3(a+b)+1\ge 0$, $3(a-b)^2- 2(a+b)\ge 0\}$; \item{(2)} $M(a,b;x,y)< I(x,y)$ for all $x,y>0$ with $x\ne y$ and $(a,b)\in \{(a,b)\in\Bbb R^2: a\ge 0$, $b\ge 0$, $3(a-b)^2- 2(a+b)\le 0\}\cup \{(a,b)\in\Bbb R^2:a+b<0\}$; \item{(3)} if $(a,b)\in\{(a,b)\in\Bbb R^2:a>0$, $b>0$, $3(a-b)^2- 2(a+b)>0\}\cup \{(a,b)\in\Bbb R^2:ab<0$, $3(a-b)^2- 2(a+b)<0\}$, then there exist $x_1,y_1,x_2,y_2>0$ such that $M(a,b;x_1,y_1)> I(x_1,y_1)$ and $M(a,b;x_2,y_2)< I(x_2,y_2)$. \par}
MSC 2000:
*26E60 Means

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