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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:
 2.6e+61 Means
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##### References:
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