设 $a_{1}, a_{2}, a_{3}$ 为正数, $E=\frac{a_{1} a_{2}}{a_{3}}+\frac{a_{2} a_{3}}{a_{1}}+\frac{a_{3} a_{1}}{a_{2}}, F=a_{1}+a_{2}+a_{3}$, 则 $E, F$ 的大小关系是
解析: 解答: 不妨设 $a_{1} \geq a_{2} \geq a_{3}>0 \quad a_{1} \geq a_{2} \geq a_{3}>0$, 于 是 $\frac{1}{a_{1}} \leq \frac{1}{a_{2}} \leq \frac{1}{a_{3}}$,

$a_{2} a_{3} \leq a_{3} a_{1} \leq a_{1} a_{2}$

<ImageHere>$rac{a_{1} a_{2}}{a_{3}}+rac{a_{3} a_{1}}{a_{2}}+rac{a_{2} a_{3}}{a_{1}} \geq rac{1}{a^{2}} \cdot a_{2} a_{3}+rac{1}{a^{3}} \cdot a_{3} a_{1}+rac{1}{a^{4}} \cdot a_{1} a_{2}=a_{3}+a_{1}+a_{2}$,即 $rac{a_{1} a_{2}}{a_{3}}+rac{a_{2} a_{3}}{a_{1}}+rac{a_{3} a_{1}}{a_{2}} \geq a_{1}+a_{2}+a_{3} rac{a_{1} a_{2}}{a_{3}}+rac{a_{2} a_{3}}{a_{1}}+rac{a_{3} a_{1}}{a_{2}} \geq a_{1}+a_{2}+a_{2} 	herefore E \geq F$分析:本题主要考查了排序不等式, 解决问题的关键是, 不妨$a_{1} \geq a_{2} \geq a_{3}>0 a_{1} \geq a_{2} \geq a_{3}>0$, 于是 $rac{1}{a_{1}} \leq rac{1}{a_{2}} \leq rac{1}{a_{3}}, a_{2} a_{3} \leq a_{3} a_{1} \leq a_{1} a_{2}$ 根据排序不等式 $a_{1} b_{1}+a_{2} b_{2}+\cdots a_{n} b_{n} \geq a_{1} c_{1}+a_{2} c_{2}+\cdots+a_{n} c_{n} \geq a_{1} b_{n}+a_{2} b_{n-1}+\cdots a_{n} b_{1}$分析计算即可.