I have been working through the Exercises at the end of Chapter $1$ of Bollobas' Linear Analysis. Chapter $1$ is on inequalities, and the text is fairly brief. I have found the problems unexpectedly hard - I guess I am simply not familiar enough with the subject. In particular, I am stuck on this one:
Let $f(x)\ge0$ be a convex function. Prove that $$\int_0^\infty\ f^2 dx\le\frac23\max f\int_0^\infty f\ dx$$ Show that $\frac{2}{3}$ is best possible.
It is easy to show that $f$ is monotone decreasing $($unless $\int_0^\infty f\ dx=\infty$ in which case the inequality is trivial$)$. It is also easy to show that if $f(0)=a,~f(b)=0$ with $f$ linear on $[0,b]$ $($and $0$ for $x>b$$)$ then we get equality, so that $\frac23$ is indeed best possible. But I still cannot hit on the right way to prove the inequality.