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Some comparison inequalities for generalized Muirhead and identric means. (English) Zbl 1187.26018
Summary: For \(x,y>0\), \(a,b\in\mathbb R\), with \(a+b\neq0\), 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 \(I(x,y)= (1/e)(y^y/x^x)^{1/(y-x)}\), \(x\neq y\), \(I(x,y)=x\), \(x=y\), respectively. In this paper, the following results are established:
(1)
\(M(a,b;x,y)> I(x,y)\) for all \(x,y>0\) with \(x\neq y\) and \((a,b)\in \{(a,b)\in\mathbb R^2: a+b>0\), \(ab\leq 0\), \(2(a-b)^2 -3(a+b)+1\geq 0\), \(3(a-b)^2- 2(a+b)\geq 0\}\);
(2)
\(M(a,b;x,y)< I(x,y)\) for all \(x,y>0\) with \(x\neq y\) and \((a,b)\in \{(a,b)\in\mathbb R^2: a\geq 0\), \(b\geq 0\), \(3(a-b)^2- 2(a+b)\leq 0\}\cup \{(a,b)\in\mathbb R^2:a+b<0\}\);
(3)
if \((a,b)\in\{(a,b)\in\mathbb R^2:a>0\), \(b>0\), \(3(a-b)^2- 2(a+b)>0\}\cup \{(a,b)\in\mathbb R^2:ab<0\), \(3(a-b)^2- 2(a+b)<0\}\), 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 \(M(a,b;x_2,y_2)< I(x_2,y_2)\).

MSC:
26E60 Means
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References:
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