设 n 为正整数, 0<yi≤xi<1, 1≤i≤n. 试证: \bexlnx1+⋯+lnxnlny1+⋯+lnyn≤√1−x11−y1+⋯+1−xn1−yn.\eex
证明: 当 n=1 时, \beex \bea &\quad \frac{\ln x_1}{\ln y_1}\leq \sqrt{\frac{1-x_1}{1-y_1}}\\ &\la \frac{\ln x_1}{\sqrt{1-x_1}}\geq \frac{\ln y_1}{\sqrt{1-y_1}}\\ &\la f(x)= \frac{\ln x}{\sqrt{1-x}}\mbox{ 是 }(0,1)\mbox{ 上的递增函数}\\ &\la f'(x)=\frac{2-2x+x\ln x}{2x(1-x)^\frac{3}{2}}\geq 0\\ &\la g(x)= 2-2x+x\ln x\geq 0\\ &\la\seddm{ g'(x)=\ln x-1\leq 0\quad\sex{0<x<1}\\ g(1)=0}. \eea \eeex
当 n≥2 时, 由 n=1 时的结论, \bexlnx1+⋯+lnxnlny1+⋯+lnyn=ln(x1⋯xn)ln(y1⋯yn)≤√1−x1⋯xn1−y1⋯yn.\eex
而仅需证明 \bex1−x1⋯xn1−y1⋯yn≤1−x11−y1+⋯+1−xn1−yn.\eex
其可用数学归纳法证明. 当 n=1 时, 结论自明. 当 n=2 时, \beex \bea \frac{1-x_1x_2}{1-y_1y_2} &<\frac{(1-x_1)+(1-x_2)}{1-y_1y_2} =\frac{1-x_1}{1-y_1y_2}+\frac{1-x_2}{1-y_1y_2}\\ &<\frac{1-x_1}{1-y_1}+\frac{1-x_2}{1-y_2}. \eea \eeex
假设当 n=k 时结论成立, 则当 n=k+1 时, \beex \bea \frac{1-x_1\cdots x_{k+1}}{1-y_1\cdots y_{k+1}} &<\frac{1-x_1}{1-y_1}+ \cdots +\frac{1-x_{k-1}}{1-y_{k-1}} +\frac{1-x_kx_{k+1}}{1-y_ky_{k+1}}\\ &\quad\sex{n=k \mbox{ 时的结论}}\\ &<\frac{1-x_1}{1-y_1}+\cdots +\frac{1-x_{k+1}}{1-y_{k+1}}\\ &\quad\sex{n=2 \mbox{ 时的结论}}. \eea \eeex
杂志目录见: http://www.cnblogs.com/zhangzujin/p/3527416.html