TITLE: Help with an inequality I've found
QUESTION [5 upvotes]: I'm trying to prove the following inequality
$$ab\leq a\xi^{-1}(a)+b\xi(b)$$ $$\\ \forall a,b>0$$ 
where $\xi$ is a strictly increasing function with $\xi(0)=0$ and $\lim_{r\rightarrow +\infty} \xi(r)=+\infty$ 
Can someone help me?

REPLY [5 votes]: Since $\xi$ is strictly increasing and $\xi(0)=0$, it is enough to show that either $\xi(b)\ge a$ or $\xi^{-1}(a)\ge b$.
Suppose that $\xi(b)<a$.  If $\xi^{-1}(a)<b$, then $a<\xi(b)$, which gives a contradiction.