$abc\leq a^p/p+b^q/q+c^r/r$

Question

Let $a,b,c>0$ and $1<p,q,r<\infty$ with $1/p+1/q+1/r=1$. Show that $abc\leq a^p/p+b^q/q+c^r/r$.

I was given a hint and it says: use Jensen's inequality.

But I don't have a clue. Jensen's inequality is about convex function and integral. How do I use it to show that?

Answer 1

It's just AM-GM: $$\frac{a^p}{p}+\frac{b^q}{q}+\frac{c^r}{r}\geq \left(a^p\right)^{\frac{1}{p}}\left(b^q\right)^{\frac{1}{q}}\left(c^r\right)^{\frac{1}{r}}=abc$$

Also we can use the cocavity of $\ln$.