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 $0 \le \rho_1, \rho_2 \in L^1$

Question

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?

Answer 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 $2C$.