Let $\Omega$ be a bounded smooth domain in $\mathbb R^2$. Then we have Ladyzhenskaya's inequality $\norm{f}_4^2 \leq c_0\norm{f}_2\norm{\grad f}_2,\quad\forall\,f\in H_0^1(\Omega).$ More generally, we have $(2)\quad\quad\norm{f}_6^3 \leq c_0\norm{f}_2\norm{\grad f}_2^2,\quad\forall\,f\in H_0^1(\Omega).$ Here $\norm{\cdot}_p$ denote the $L^p$-norm in $\Omega$. The inequality $(2)$ follows from $(3)\quad\quad\norm{f}_6^6 \leq \norm{f}_8^{16/3} \norm{f}_2^{2/3},\quad\quad \norm{f}_8^8 \leq \norm{f}_6^6 \norm{\grad f}_2^2.$ See [[http://math.7starsea.com/post/246|Agmon's inequality]] for a proof of the second inequality in $(3)$. The first inequality in $(3)$ is a direct application of Holder's inequality.