Given $ \Omega$ a compact subset of $\mathbb{R}^n$ and $f\in H^1(\Omega,\mathbb{C})$ with zero average, I wonder if there is an inequality of the form $$ \int_\Omega \phi(|f|^2)\varphi(|f|^2)\ dx \leq C\left( \int_\Omega \phi(|f|^2)\ dx \right)^{1/a} \left(\int_\Omega \varphi(|f|^2)\ dx \right)^{1/a} $$ where $a>1$, $C>0$ (might depend on $\Omega$) and $\varphi,\phi$ are real-valued and convex functions. Any idea is welcome, thanks for advance.

I checked this for the simple case of $\varphi,\phi$ being constants and it works (it is enough to take $C\geq 1$).