Integral inequality with f(x)/x and f'(x)

$\mathbf{Problem:}$ Suppose $f\in C^1[0,a]$, and $f(0)=0$, then

$\displaystyle \int_0^a\left(\frac{f(x)}{x}\right)^2dx\leq 4\int_0^a|f'(x)|^2dx$

$\mathbf{Proof:}$ From integration by parts, we get

$\displaystyle \int_\epsilon^a f^2(x)\left(\frac{-1}{x}\right)'dx=\frac{-1}{x}f^2(x)\vert_\epsilon^a+\int_0^a 2f'(x)\frac{f(x)}{x}dx$

letting $\epsilon \to 0$, note the fact that $\displaystyle \lim\limits_{x\to 0}\frac{f^2(x)}{x}=0$, we get

$\displaystyle \int_0^a \left(\frac{f(x)}{x}\right)^2dx=2\int_0^a f'(x)\frac{f(x)}{x}dx-\frac{f^2(a)}{a}$

So by Cauchy’s inequality

$\displaystyle \int_0^a \left(\frac{f(x)}{x}\right)^2dx\leq 2\int_0^a f'(x)\frac{f(x)}{x}dx\leq 2 \int_0^a \left(f'(x)\right)^2dx+\frac{1}{2}\int_0^a\left(\frac{f(x)}{x}\right)^2dx$

The inequality follows from this easily.

$\text{Q.E.D}\hfill \square$

$\mathbf{Remark:}$ I first saw this from Existence of solutions to the nonhomogeneous steady Navier-Stokes Equations. by Charles, J. Amick