An application of Holder's inequality to show one norm is smaller than another

Question

Let $p(s) = r(s) + m-1$ where $r:[0,T) \to [q,\infty)$ where $q \geq 2$ and $m > 1$ is fixed.

Let $\text{Vol}(\Omega) = 1$.

Then can we show that $$\lVert u \rVert_{L^{r(s)}(\Omega)} \leq \lVert u \rVert_{L^{p(s)}(\Omega)}?$$ I know I need to use Holder's inequality but I cannot do it.

Answer 1

One can forget $p(s)$, $r(s)$ and the rest and simply try to show that, for every $a\lt b$, $$\|u\|_a\leqslant\|u\|_b.$$ To wit, considering $v=|u|^a$ and $p=b/a\gt1$, note that Hölder inequality yields $$ \int |u|^a=\int v\leqslant\left(\int v^p\right)^{1/p}=\left(\int |u|^b\right)^{a/b}, $$ that is, $$ \left(\int |u|^a\right)^{1/a}\leqslant\left(\int |u|^b\right)^{1/b}, $$