How to prove $|\Omega|^{-1/p}||u||_p\leq |\Omega|^{-1/q}||u||_q $?

Question

How to prove $|\Omega|^{-1/p}||u||_p\leq |\Omega|^{-1/q}||u||_q $?

$u\in L^q(\Omega)$.

I guess using Holder inequality above inequality is true.

But I could not properly arrange term to get required?

Please give me a hint so that I can prove the above inequality?

Any Help/ Hint will be appreciated.

Answer 1

This is true only for $0<p<q$. By Holders' inequality $\int |u|^{p} =\int (1) (|u|^{p})\leq (\int 1^{r})^{1/r} (\int |u|^{q})^{p/q} =|\Omega|^{r} (\int |u|^{q})^{p/q}$ where $\frac 1 r +\frac p q=1$. Hence $(\int |u|^{p})^{1/p} \leq |\Omega|^{1/rp} (\int |u|^{q})^{1/q}$. Just compute $r$ to finish the proof.