Estimating $\int_0^x f(x-t)f'(t)dt$

Question

I'm attempting to estimate $\int_0^x f(x-t)f'(t)dt$ in terms of a simple asymptotic expression with an error term for some 'well-behaved' functions, namely $f = O(x)$, of class $C^1$ or higher, with $f$ (asymptotically) increasing and $f'$ (asymptotically) monotonic. (e.g. $\sqrt[3]{x}$, $\frac{x}{\log{x}}$, $\sqrt{x}\log{x}$).

If $f$ is not positive or well-defined in $(0,x)$ we can just change the domain of integration to $(\alpha, x-\alpha)$, this change doesn't modify any asymptotic result anyway.

Example. For $f(x) = \sqrt{x}$, it's not hard to show that $\int_0^x f(x-t)f'(t)dt \sim \frac{\pi}{4}x$.

More generally, I suspect that $\int_0^x f(x-t)f'(t)dt = Cf(x)^2 + O(f(x))$, for some constant $C$ depending on $f$, but the only thing that I have sketched a proof so far was that $\int_0^x f(x-t)f'(t)dt \asymp f(x)^2$, as follows:

Firstly, by the order of $f$, holds $f(x-t) \gg f(x)-f(t)$ as $x\to\infty$. So:

$\int_0^x f(x-t)f'(t)dt \gg f(x)\int_0^x f'(t)dt - \int_0^x f(t)f'(t)dt \gg f(x)^2$.

Secondly, given that $f$ is increasing,

$\int_0^x f(x-t)f'(t)dt < f(x)\int_0^{x/2} f'(t)dt + \int_{x/2}^x f(t)f'(t)dt \ll f(x)^2$.

The problem is that I have no clue how to prove (or if I have to add more hypothesis over $f$ in order to prove) my suspicion. I would appreciate any suggestion.

Answer 1

To simplify things, I assume that $f$ is increasing and $f'$ is decreasing* on $(0,\infty)$; otherwise shift the variable. Begin with $$\begin{split} \int_0^x f(x-t)f'(t)\,dt &= \int_0^x \left(f(0)+\int_0^{x-t}f'(s)\,ds\right)f'(t)\,dt \\&=f(0)f(x) + \iint_{s+t\le x}f'(s)f'(t)\,ds\,dt \end{split}$$ (The variables of integration are always nonnegative here).

Also, $f(x)^2= -f(0)^2+2f(0)f(x)+(f(x)-f(0))^2$ where the last term is $$ (f(x)-f(0))^2 = \left(\int_0^{x}f'(t)\,dt\right)^2 = \iint_{s,t\le s}f'(s)f'(t)\,ds\,dt $$ Since $f'$ is nonnegative and $\{(s,t):s+t\le x\}\subset \{(s,t):s,t\le x\}$, it follows that $$ \int_0^x f(x-t)f'(t)\,dt \le f(x)^2 + O(f(x)) \tag{1} $$ On the other hand, $\{(s,t):s+t\le x\}\supset \{(s,t):s,t\le x/2\}$, which leads to $$ \int_0^x f(x-t)f'(t)\,dt \ge f(x/2)^2 + O(f(x)) \tag{2} $$ And since $f'$ is decreasing*, $f(x)-f(x/2)\le f(x/2)-f(0)$, which implies $f(x)\le 2f(x/2)+f(0)$, hence $f(x)^2\le 4f(x/2)^2 + O(f(x))$. So, (1) and (2) imply $$ \frac14 f(x)^2 + O(f(x)) \le \int_0^x f(x-t)f'(t)\,dt \le f(x)^2 + O(f(x)) \tag{3} $$

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I don't believe that the limit $\frac{1}{f(x)^2}\int_0^x f(x-t)f'(t)\,dt $ exists without additional assumptions on $f$. The problem is that $f'$ may decrease somewhat irregularly, so that the integrals of $f'(s)f'(t)$ over a square and its left-lower diagonal half don't have to be asymptotically proportional.

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(*) You did not say that $f'$ is decreasing, only that it's monotone. But if $f'$ is increasing, then due to the assumption $f(x)=O(x)$ we have $f'(x)\to L<\infty$, hence $f(x)\sim Lx$, from where the conclusion $\int_0^x f(x-t)f'(t)\,dt\sim \frac12 L^2 x^2\sim \frac12 f(x)^2$ follows.