Find all functions such that $ (\int _0 ^1 xf(x) dx)^3 = \frac{4}{25} \int _0 ^1 f(x)^3 dx$

Question

Calculate all the functions $f \in L^3$ such that

$$ \left(\int _0 ^1 xf(x) dx\right)^3 = \frac{4}{25} \int _0 ^1 f(x)^3 dx$$

Can someone please walk me through this because there are no such examples in the book.

Answer 1

Using the Holder inequality, we have $$ \left(\int_0^1 |x\,f(x)|\,dx\right)^3 \leq \left(\int_0^1 x^{3/2}\,dx\right)^2 \left(\int_0^1 |f(x)|^3\,dx\right) = \frac{4}{25} \int_0^1 |f(x)|^3\,dx. $$ But from our equation above we know this must be an equality. The Holder inequality is sharp exactly when one function is a constant multiple of the other, so we conclude that $f(x) = cx$ for some $c\in\mathbb{R}$.