Denote by $L^1(0,1)$ the space of Lebesgue integrable functions on the interval $(0,1)$.

$\textbf{Question:}$ Does there exist a function $F:(0,1)\rightarrow\mathbb{R}$ such that:

1. $\frac{F(x)}{x}\in L^1(0,1)$,
2. $\frac{F'(x)}{x}\in L^1(0,1)$,
3. $\frac{F(x)}{x^2}\notin L^1(0,1)$?

I'm guessing that the answer is positive and the point is to construct $F$ such that $F$ and $F'$ behave suitably near zero. It seems quite delicate. I checked that $F$ cannot be a polynomial or a power function (since then $F'\simeq \frac{F}x$, thus conditions 2 and 3 cannot hold simultaneously).

I would appreciate any hints!