Is the $||u||L_{(U)}^p$ greater than the average of u over U?

Question

Is it true that for any function u, the Lp-norm, $||u||L_{(U)}^p$ is always greater than or equal to the average value of u over U for all p?

Answer 1

If $u= 1$, $p<∞$, and $U=[-1/10,1/10]$ then the average value is $1$,with $$‖u‖_{L^p(U)} = (1/5)^{1/p} < 1 $$

For $p=∞$, $\frac{1}{|U|}\left|\int_U u\right| ≤ ‖u‖_{L^∞}$ just by Holder's inequality. (In fact Holder lets you get more counterexamples for the $p<∞$ case.)