I am interested in the following extremal-type problem.
Let us define $\Psi$ by $$\Psi(x)=\max_{f\in L^2[0,x] \,\,\text{with}\,\,\|f\|_2=1}\Bigg|\int_0^x\int_0^xf(t)f(s)\ln|t-s|dsdt\Bigg|$$ on $(0,\infty)$. Is $\Psi$ injective? In particular can it be represented explicitly?(perhaps $\Psi(x)$ there is some connections with logarithmic capacity of a segment of length $x$)
$\textbf{Edit 1:}$ It is obvious that if $x<y$ then $\Psi(x)\leq\Psi(y)$, what is not apparent is that whether the inequality is strict or not.
$\textbf{Edit 2:}$ Is the maximum attained?Or should it be replaced with supremum?