【微分不等式】

Mark

f(x)=∫(0 to x)f'(x)dx
f(x)=∫(1 to x)f'(x)dx
|f(x)|=0.5|∫(0 to x)f'(x)dx|+0.5|∫(1 to x)f'(x)dx|
≤0.5∫(0 to x)|f'(x)|dx+0.5∫(1 to x)|f'(x)|dx
=0.5∫(0 to 1)|f'(x)|dx
即有：|f(x)|≤0.5∫(0 to 1)|f'(x)|dx
同时平方，有：
f²(x)≤0.25(∫(0 to 1)|f'(x)|dx)²≤0.25∫(0 to 1)|f'(x)|²dx
两边同时对x在(0,1)区间积分即得证。