Let $\rho\_1, \rho\_2 \in L^1(\Omega;\mathbb R\_+)$ such that $\int \rho\_i|\ln \rho\_i| < \infty$. Is it true that there exists a constant $C>0$ such that
\begin{align\*}
\int\_\Omega \left(\rho\_{1} \ln \frac{\rho\_{1}}{\rho\_{2}}\right) d x d y \leq C\int\_\Omega |\rho\_1 - \rho\_2| d x d y 
\end{align\*}
holds?



Prove $\int_\Omega \left(\rho_{1} \ln \frac{\rho_{1}}{\rho_{2}}\right)dx dy \leq C\int_\Omega |\rho_1-\rho_2|dxdy$ for $0 \le \rho_1, \rho_2 \in L^1$
Certainly not, if $C$ is supposed to be real. For instance, suppose that $\rho\_1$ and $\rho\_2$ are probability densities such that $\rho\_1\rho\_2=0$. Then the left-hand side of your inequality is $\infty$, whereas its right-hand side is $2C$.


