I have been recently trying to prove a lemma which seems true in every single example I have tried, yet that I didn't manage to prove so far unless making extra (not desirable) assumptions.
A formulation of the question is a follows*: fix $L \geq 2$. Set $I=[1,L]$; and let $f\colon I \to [0,1]$ be a monotone (non-increasing) integrable function. Now, for $A_f\stackrel{\rm{}def}{=} \int_I f > 0$, I would like to argue something of the form $$ \int_I \left( f-\frac{A_f}{L}\right)^2 \leq c\cdot\frac{A_f^2}{L} \tag{$\dagger$} $$ for some absolute constant $c$, ideally $c=1$. (or, in the worst case, the best bound one could get on the RHS, involving $1/L$.)
I can prove $(\dagger)$, under the assumption that the ratio $f(1)/f(L)$ is bounded by a constant; but without this assumption, I'm at a loss.
*Hopefully, I didn't screw up in the reformulation -- my original problem is a bit less easy to state, and for discrete functions $f\colon\mathbb{N}\to[0,1]$.
If $f(x)=\begin{cases} 1 & 1\le x<1+\varepsilon \\ 0 & 1+\varepsilon \le x \ge L \\ \end{cases}$ then the LHS is $\varepsilon+O(\varepsilon^2)$ and the RHS is $O(\varepsilon^2)$, so such $c$ does not seem to exist.