Assume $P_1,P_2,P_3$ pmfs. We would like to find an upper bound for $D_{\text{KL}}(P_1\ast P_3||P_2 \ast P_3)$, where $D_{KL}$ is the [Kullback-Leibler][1] divergence and $*$ is convolution. By using the log sum inequality we can get $D_{\text{KL}}(P_1\ast P_3||P_2 \ast P_3) \leq D_{\text{KL}}(P_1||P_2 )$. Is there any way to get a tighter bound? 


  [1]: https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence