谢惠民上册的一道不等式题

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谢上P9的一道不等式题，以前写过，但文件丢失，先前的解答难以回忆起，现在重新给出解答。

用向前-向后数学归纳法证明:设$\displaystyle 0<x_i\leq \frac12,i=1,2,\cdots,n$,则

\[\frac{{\prod\limits_{i = 1}^n {{x_i}} }}{{{{\left( {\sum\limits_{i = 1}^n {{x_i}} } \right)}^n}}} \le \frac{{\prod\limits_{i = 1}^n {\left( {1 - {x_i}} \right)} }}{{{{\left[ {\sum\limits_{i = 1}^n {\left( {1 - {x_i}} \right)} } \right]}^n}}}.\]

(这个不等式是由在美国数学界有重大影响的华裔数学家樊畿(Fan Ky)得到的,关于它的许多研究和推广见[30].)

首先,

\[\frac{{{{\left[ {\sum\limits_{i = 1}^n {\left( {1 - {x_i}} \right)} } \right]}^n}}}{{{{\left( {\sum\limits_{i = 1}^n {{x_i}} } \right)}^n}}} \le \frac{{\prod\limits_{i = 1}^n {\left( {1 - {x_i}} \right)} }}{{\prod\limits_{i = 1}^n {{x_i}} }} \Leftrightarrow {\left[ {\frac{n}{{\sum\limits_{i = 1}^n {{x_i}} }} - 1} \right]^n} \le \prod\limits_{i = 1}^n {\left( {\frac{1}{{{x_i}}} - 1} \right)} .\]

当$n=2$时,即证

\[\left( {\frac{1}{{{x_1}}} - 1} \right)\left( {\frac{1}{{{x_2}}} - 1} \right) \ge {\left[ {\frac{2}{{{x_1} + {x_2}}} - 1} \right]^2}.\]

展开得

\[\frac{1}{{{x_1}{x_2}}} - \frac{1}{{{x_1}}} - \frac{1}{{{x_2}}} + 1 \ge \frac{4}{{{{\left( {{x_1} + {x_2}} \right)}^2}}} - \frac{4}{{{x_1} + {x_2}}} + 1.\]

等价于证明

\[\frac{1}{{{x_1}{x_2}}} - \frac{4}{{{{\left( {{x_1} + {x_2}} \right)}^2}}} \ge \frac{1}{{{x_1}}} + \frac{1}{{{x_2}}} - \frac{4}{{{x_1} + {x_2}}} \Leftrightarrow \frac{{{{\left( {{x_1} - {x_2}} \right)}^2}}}{{{{\left( {{x_1} + {x_2}} \right)}^2}{x_1}{x_2}}} \ge \frac{{{{\left( {{x_1} - {x_2}} \right)}^2}}}{{\left( {{x_1} + {x_2}} \right){x_1}{x_2}}}.\]

注意到$x_1+x_2\leq 1$,上式显然成立.

从$n=2$的已知情况出发,可以得到如下$n=4$时的情形:

\begin{align*}&\prod\limits_{i = 1}^4 {\left( {\frac{1}{{{x_i}}} - 1} \right)} = \prod\limits_{i = 1}^2 {\left( {\frac{1}{{{x_i}}} - 1} \right)} \cdot \prod\limits_{i = 3}^4 {\left( {\frac{1}{{{x_i}}} - 1} \right)} \ge {\left[ {\frac{2}{{\sum\limits_{i = 1}^2 {{x_i}} }} - 1} \right]^2}{\left[ {\frac{2}{{\sum\limits_{i = 3}^4 {{x_i}} }} - 1} \right]^2}\\= &{\left[ {\left( {\frac{1}{{\frac{1}{2}\sum\limits_{i = 1}^2 {{x_i}} }} - 1} \right)\left( {\frac{1}{{\frac{1}{2}\sum\limits_{i = 3}^4 {{x_i}} }} - 1} \right)} \right]^2} \le {\left[ {{{\left( {\frac{2}{{\frac{1}{2}\sum\limits_{i = 1}^2 {{x_i}} + \frac{1}{2}\sum\limits_{i = 3}^4 {{x_i}} }} - 1} \right)}^2}} \right]^2} = {\left[ {\frac{4}{{\sum\limits_{i = 1}^4 {{x_i}} }} - 1} \right]^4}.\end{align*}

同样可知,若$n=2^k$时不等式已成立,则可得到$n=2^{k+1}$时的不等式

\begin{align*}&\prod\limits_{i = 1}^{{2^{k + 1}}} {\left( {\frac{1}{{{x_i}}} - 1} \right)} = \prod\limits_{i = 1}^{{2^k}} {\left( {\frac{1}{{{x_i}}} - 1} \right)} \cdot \prod\limits_{i = {2^k} + 1}^{{2^{k + 1}}} {\left( {\frac{1}{{{x_i}}} - 1} \right)} \ge {\left[ {\frac{{{2^k}}}{{\sum\limits_{i = 1}^{{2^k}} {{x_i}} }} - 1} \right]^{{2^k}}}{\left[ {\frac{{{2^k}}}{{\sum\limits_{i = {2^k} + 1}^{{2^{k + 1}}} {{x_i}} }} - 1} \right]^{{2^k}}}\\= & {\left[ {\left( {\frac{1}{{\frac{1}{{{2^k}}}\sum\limits_{i = 1}^{{2^k}} {{x_i}} }} - 1} \right)\left( {\frac{1}{{\frac{1}{{{2^k}}}\sum\limits_{i = {2^k} + 1}^{{2^{k + 1}}} {{x_i}} }} - 1} \right)} \right]^{{2^k}}} \le {\left[ {{{\left( {\frac{2}{{\frac{1}{{{2^k}}}\sum\limits_{i = 1}^{{2^k}} {{x_i}} + \frac{1}{{{2^k}}}\sum\limits_{i = {2^k} + 1}^{{2^{k + 1}}} {{x_i}} }} - 1} \right)}^2}} \right]^{{2^k}}} = {\left[ {\frac{{{2^{k + 1}}}}{{\sum\limits_{i = 1}^{{2^{k + 1}}} {{x_i}} }} - 1} \right]^{{2^{k + 1}}}}.\end{align*}

这样就证明了当$n$为$2$的所有方幂时平均值不等式已成立.这是“向前”部分.

第二步要证明,当平均值不等式对某个$n>2$成立时,则它对$n-1$也一定成立.这是证明中的“向后”部分.写出

\begin{align*}&{\left[ {\frac{{n - 1}}{{\sum\limits_{i = 1}^{n - 1} {{x_i}} }} - 1} \right]^{n - 1}} = {\left[ {\frac{n}{{\sum\limits_{i = 1}^{n - 1} {{x_i}} + \frac{1}{{n - 1}}\sum\limits_{i = 1}^{n - 1} {{x_i}} }} - 1} \right]^n} \cdot {\left[ {\frac{{n - 1}}{{\sum\limits_{i = 1}^{n - 1} {{x_i}} }} - 1} \right]^{ - 1}}\\\le &\prod\limits_{i = 1}^{n - 1} {\left( {\frac{1}{{{x_i}}} - 1} \right)} \cdot \left( {\frac{1}{{\frac{1}{{n - 1}}\sum\limits_{i = 1}^{n - 1} {{x_i}} }} - 1} \right) \cdot {\left[ {\frac{{n - 1}}{{\sum\limits_{i = 1}^{n - 1} {{x_i}} }} - 1} \right]^{ - 1}} = \prod\limits_{i = 1}^{n - 1} {\left( {\frac{1}{{{x_i}}} - 1} \right)} .\end{align*}

于是$n-1$时不等式也成立.合并以上向前和向后两部分,可见不等式对每个自然数$n$成立.

事实上,我们有

\[{\left[ {\frac{n}{{\sum\limits_{i = 1}^n {{x_i}} }} - 1} \right]^n} \le \prod\limits_{i = 1}^n {\left( {\frac{1}{{{x_i}}} - 1} \right)}  \Leftrightarrow \frac{1}{n}\sum\limits_{i = 1}^n {\ln \left( {\frac{1}{{{x_i}}} - 1} \right)}  \ge \ln \left( {\frac{1}{{\frac{1}{n}\sum\limits_{i = 1}^n {{x_i}} }} - 1} \right).\]结合函数$y = \ln \left( {\frac{1}{x} - 1} \right)$的凹凸性便可得证.