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&gt;b$$)$ then we get equality, so that $\frac23$ is indeed best possible. But I still cannot hit on the right way to prove the inequality.
Prove $$\int _0^\infty f^2(x) dx \leq \frac{2}{3}\cdot  \max_{x \in \mathbb R^+} f(x) \cdot  \int _0^\infty f(x) dx$$ for $f(x) \geq 0$ and convex.   I know via Holder's we can get without the constant $\frac{2}{3}$ but that should also be true for $f$ not convex. I tried some approaches using convexity but didn't get the desired result.   A hint or reference would also help. Also, it would be interesting to know when equality holds.
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