Proving $||f||^2_{L^2(\mathbb R^2)}\le 10||f||_{L^1(\mathbb R^2)}||\nabla f||_{L^2(\mathbb R^2)}$ with $f\in C^1,L^1\cap L^2$ and $\nabla f\in L^2$.

Question

Prove that there exists a universal constant $K<10$, for all $C^1$ function $f : \mathbb R^2 \rightarrow\mathbb R$, if $f \in L^1 (\mathbb R^2)\cap L^2(\mathbb R^2)$ and $|\nabla f| \in L^2(\mathbb R^2)$, we have the following inequality: $$||f||^2_{L^2(\mathbb R^2)}\le K||f||_{L^1(\mathbb R^2)}||\nabla f||_{L^2(\mathbb R^2)}$$ $$$$ Since this form is something like "A function's norm can be bounded by its gradient's norm", and the problem provides many limitations on $f$ and $\nabla f$, I think that this problem should be solved by Poincaré inequality since the forms are too similar. However there're two problems: First the Poincaré inequality is used in a bounded domain, second the Poincaré inequality didn't provide any explicit constants. My question is, can this problem be solved by using Poincaré inequality (or relevant inequalities)? What is it like if it can? If it can't, then what ways should we try? Thanks!