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

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