According to the pinsker inequality, we have the following inequality: \begin{equation} \delta_{TV} (p, q)^2 \leq \frac{1}{2} D_{KL}(p,q), \end{equation} where $\delta_{TV} (\cdot, \cdot)$ and $D_{KL}(\cdot, \cdot)$ are total variation distance and Kullback–Leibler divergence, respectively.

On the other hand, the total variation distance is related to the $L_1$ norm by the identity: \begin{equation} \delta_{TV}(p, q) = \frac{1}{2} \int_{\mathcal{X}} |p(x)-q(x)| \, dx, \end{equation} and thus by using the Cauchy–Schwarz inequality we obtain that \begin{equation} \delta_{TV} (p, q)^2 \leq \frac{1}{4} \int_{\mathcal{X}} (p(x)-q(x))^2 \, dx. \end{equation} I denote the RHS by $L_2(p, q)$, i.e., $L_2(p, q) = \int_{\mathcal{X}} (p(x)-q(x))^2 \, dx$.

My question is: does there exist some (inequality) relation between the $D_{KL}(\cdot, \cdot)$ and $L_2(p, q)$?

According to the Pinsker inequality, we have the following inequality: \begin{equation} \delta_{TV} (p, q)^2 \leq \frac{1}{2} D_{KL}(p,q), \end{equation} where $\delta_{TV} (\cdot, \cdot)$ and $D_{KL}(\cdot, \cdot)$ are total variation distance and Kullback–Leibler divergence, respectively.

On the other hand, the total variation distance is related to the $L_1$ norm by the identity: \begin{equation} \delta_{TV}(p, q) = \frac{1}{2} \int_{\mathcal{X}} |p(x)-q(x)| \, dx, \end{equation} and thus by using the Cauchy–Schwarz inequality we obtain that \begin{equation} \delta_{TV} (p, q)^2 \leq \frac{1}{4} \int_{\mathcal{X}} (p(x)-q(x))^2 \, dx. \end{equation} I denote the RHS by $L_2(p, q)$, i.e., $L_2(p, q) = \int_{\mathcal{X}} (p(x)-q(x))^2 \, dx$.

My question is: does there exist some (inequality) relation between the $D_{KL}(\cdot, \cdot)$ and $L_2(p, q)$?

According to the pinsker inequality, we have the following inequality: \begin{equation} \delta_{TV} (p, q)^2 \leq \frac{1}{2} D_{KL}(p,q), \end{equation} where $\delta_{TV} (\cdot, \cdot)$ and $D_{KL}(\cdot, \cdot)$ are total variation distance and Kullback–Leibler divergence, respectively.

On the other hands, the total variation distance is related to the $L_1$ norm by the identity: \begin{equation} \delta_{TV}(p, q) = \frac{1}{2} \int_{\mathcal{X}} |p(x)-q(x)| \, dx, \end{equation} and thus by using the Cauchy–Schwarz inequality we can obtain that \begin{equation} \delta_{TV} (p, q)^2 \leq \frac{1}{4} \int_{\mathcal{X}} (p(x)-q(x))^2 \, dx. \end{equation} I denotes the RHS by $L_2(p, q)$, i.e., $L_2(p, q) = \int_{\mathcal{X}} (p(x)-q(x))^2 \, dx$.

My question is that is there exists some (inequality) relation between the $D_{KL}(\cdot, \cdot)$ and $L_2(p, q)$?

According to the pinsker inequality, we have the following inequality: \begin{equation} \delta_{TV} (p, q)^2 \leq \frac{1}{2} D_{KL}(p,q), \end{equation} where $\delta_{TV} (\cdot, \cdot)$ and $D_{KL}(\cdot, \cdot)$ are total variation distance and Kullback–Leibler divergence, respectively.

On the other hand, the total variation distance is related to the $L_1$ norm by the identity: \begin{equation} \delta_{TV}(p, q) = \frac{1}{2} \int_{\mathcal{X}} |p(x)-q(x)| \, dx, \end{equation} and thus by using the Cauchy–Schwarz inequality we obtain that \begin{equation} \delta_{TV} (p, q)^2 \leq \frac{1}{4} \int_{\mathcal{X}} (p(x)-q(x))^2 \, dx. \end{equation} I denote the RHS by $L_2(p, q)$, i.e., $L_2(p, q) = \int_{\mathcal{X}} (p(x)-q(x))^2 \, dx$.

My question is: does there exist some (inequality) relation between the $D_{KL}(\cdot, \cdot)$ and $L_2(p, q)$?