I am following the proof of the following Poincaré's inequality from the Lieb and Loss textbook
8.12 Let $Ω∈\mathbb{R}^n$ be a bounded, connected open set that has the cone property for some $θ$ and $r.$ Let $1≤p≤∞$, and let $g$ be a function in $L^{p′}(Ω)$, s.t. $\int_{Ω}g=1.$ Let $1 < q < np/(n - p)$ when $p < n,$ $q <\infty$, when $p = n,$ and $1 < q <\infty$ when $p > n.$ Then there is a finite number $S>0,$ which depends on $Ω,g,p,q$ such that $$||f-\int_{\Omega}fg|\rvert_{L^q(\Omega)}\leq S||\nabla f|\rvert_{L^p(\Omega)} $$ $∀f∈W^{1,p}(Ω).$
They begin the proof by assuming $q\geq p$. The explanation for this assumption is given as follows:
If $q < p$ we can first prove the theorem for $q = p$ and then use the fact that $\Omega$ is bounded and that the $p$ norm dominates the $q$ norm by Hölder's inequality.
What do they mean by this, does the Theorem hold in the case when $q<p$? The Hölder's inequality states $$\int_{\Omega}fh=||f|\rvert_{L^p}||h|\rvert_{L^q}, \quad {\rm{for}}\quad \frac{1}{p}+\frac{1}{q}=1$$ but I don't see how the $p$ norm dominates the $q$ norm in their argument. If anyone can please clarify on this I would really appreciate it!