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Iosif Pinelis
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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 $2C$.

Iosif Pinelis
  • 127.7k
  • 8
  • 107
  • 229