Let $0<a<b<1$. Does the following inequality hold:

$$\max_{f\in L^2[0,a],\,\,\|f\|_2=1}\Bigg|\int_0^a\int_0^af(x)f(y)\ln|x-y|dxdy\Bigg|$$ $$<\max_{g\in L^2[0,b],\,\,\|g\|_2=1}\Bigg|\int_0^b\int_0^bg(x)g(y)\ln|x-y|dxdy\Bigg|$$


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    $\begingroup$ Before any work on this logarithmic kernel, what can be said about the convergence of these double integrals ? $\endgroup$
    – Jean Marie
    Apr 20, 2016 at 21:50
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    $\begingroup$ Doesn't my Answer to your other Question apply here? $\endgroup$ Apr 21, 2016 at 4:13
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    $\begingroup$ @KeithMcClary The difficult part is to show that the inequality is strict.Your arguement to the more general case (namely my first question) merely shows the inequality: LHS $\leq$ RHS. $\endgroup$
    – BigM
    Apr 21, 2016 at 4:42
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    $\begingroup$ @WilliamKrinsman Intuitively it should be. It would be very odd if for some $a<b$, the two quantities are equal. But for some reason I cannot come up with a proof. $\endgroup$
    – BigM
    Apr 22, 2016 at 23:28
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    $\begingroup$ @BigM oh yeah I guess that idea wouldn't be sufficient if the maximum of another subclass was greater and for that other subclass only non-strict inequality holds. shoot. $\endgroup$ Apr 22, 2016 at 23:30


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