We know the general version of Poincare inequality:
$$ \int_\Omega |u-u_\Omega|^2dx\leq C\int_\Omega|\nabla u|^2dx,\quad \forall u\in W^{1,2}(\Omega), $$ where $u_\Omega$ is the average of $u$ over $\Omega$. This immediately shows that $$ \int_\Omega |u|^2dx\leq C\int_\Omega|\nabla u|^2dx+\frac{1}{|\Omega|}\Bigr(\int_\Omega udx\Bigr)^2,\quad \forall u\in W^{1,2}(\Omega). $$
Now, the question is how to deduce the following modified Poincare inequality: for any $m\geq2$, there holds $$ \int_\Omega |u|^2dx\leq C\Bigr\{\int_\Omega|\nabla u|^2dx+\Bigr(\int_\Omega |u|^{\frac{2}{m}}dx\Bigr)^m\Bigr\},\quad \forall u\in W^{1,2}(\Omega), $$ where $C>0$ depends only on $m$ and $\Omega$. My tentative way is to use interpolation by means of the Holder inequlity to the second inequality above. But I can not get the desired result. Any help is highly appreciated.