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Some comparison inequalities for generalized Muirhead and identric means. (English) Zbl 1187.26018

Summary: For $x,y>0$, $a,b\in ℝ$, with $a+b\ne 0$, the generalized Muirhead mean $M\left(a,b;x,y\right)$ with parameters $a$ and $b$ and the identric mean $I\left(x,y\right)$ are defined by $M\left(a,b;x,y\right)={\left(\left({x}^{a}{y}^{b}+{x}^{b}{y}^{a}\right)/2\right)}^{1/\left(a+b\right)}$ and $I\left(x,y\right)=\left(1/e\right){\left({y}^{y}/{x}^{x}\right)}^{1/\left(y-x\right)}$, $x\ne y$, $I\left(x,y\right)=x$, $x=y$, respectively. In this paper, the following results are established:

$M\left(a,b;x,y\right)>I\left(x,y\right)$ for all $x,y>0$ with $x\ne y$ and $\left(a,b\right)\in \left\{\left(a,b\right)\in {ℝ}^{2}:a+b>0$, $ab\le 0$, $2{\left(a-b\right)}^{2}-3\left(a+b\right)+1\ge 0$, $3{\left(a-b\right)}^{2}-2\left(a+b\right)\ge 0\right\}$;

$M\left(a,b;x,y\right) for all $x,y>0$ with $x\ne y$ and $\left(a,b\right)\in \left\{\left(a,b\right)\in {ℝ}^{2}:a\ge 0$, $b\ge 0$, $3{\left(a-b\right)}^{2}-2\left(a+b\right)\le 0\right\}\cup \left\{\left(a,b\right)\in {ℝ}^{2}:a+b<0\right\}$;

if $\left(a,b\right)\in \left\{\left(a,b\right)\in {ℝ}^{2}:a>0$, $b>0$, $3{\left(a-b\right)}^{2}-2\left(a+b\right)>0\right\}\cup \left\{\left(a,b\right)\in {ℝ}^{2}:ab<0$, $3{\left(a-b\right)}^{2}-2\left(a+b\right)<0\right\}$, then there exist ${x}_{1},{y}_{1},{x}_{2},{y}_{2}>0$ such that $M\left(a,b;{x}_{1},{y}_{1}\right)>I\left({x}_{1},{y}_{1}\right)$ and $M\left(a,b;{x}_{2},{y}_{2}\right).

##### MSC:
 2.6e+61 Means