multiple integration question

May 2010
1
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If f(x,y) is nonnegative decrasing in x for fixed y, 0<x<1, 0<y<0.4 say, then one can show that the quadruple integral

\(\displaystyle
\int_{0}^{u} \int_{u}^{1} \int_{0}^{0.4} \int_{0}^{0.4} (f(x,y)-f(x',y')) dy dy' dx dx' \leq 0 \).

Is it true that

\(\displaystyle
\int_{0}^{u} \int_{u}^{1} \int_{0}^{0.4} \int_{0}^{0.4} g(x,y) g(x',y') [f(x,y)-f(x',y')] dy dy' dx dx' \leq 0 \).


for any non-negative function g(x,y) ? I think it is but can't seem to come up with a simple proof without going through the process of simple functions etc. Other assumptions are

f and g are bounded by 1, continously differentiable with respect to each parameter.

Thanks,
 
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