# Strict inequality for logarithmic integrals

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

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

?

• Before any work on this logarithmic kernel, what can be said about the convergence of these double integrals ? Apr 20, 2016 at 21:50
• Doesn't my Answer to your other Question apply here? Apr 21, 2016 at 4:13
• @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.
– BigM
Apr 21, 2016 at 4:42
• @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.
– BigM
Apr 22, 2016 at 23:28
• @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. Apr 22, 2016 at 23:30