Let $\Omega$ be a countable set and $\mu,\nu\colon\Omega\to[0,1]$ be distributions on $\Omega$, that is we have $\sum_{x\in\Omega}\mu(x)=1$ and likewise for $\nu$. The Kullback-Leibler divergence of $\mu$ from $\nu$ is given by \begin{align*} {\mathbf D}(\mu\,\|\,\nu):=\sum_{x\in\Omega}\mu(x)\log\frac{\mu(x)}{\nu(x)}. \end{align*}

I am interested in lower bounding quantities of the form ${\mathbf D}(\mu\,\|\,\epsilon\mu+(1-\epsilon)\nu)$ for some $\epsilon\in[0,1]$. What are the keywords to search for such expressions in the literature? I was guessing 'hypothesis testing' would take me there, but to no avail so far.

I am guessing something like the following should hold \begin{align*} {\mathbf D}(\mu\,\|\,\epsilon\mu+(1-\epsilon)\nu) \geq \frac{1}{2} {\mathbf D}(\mu\,\|\,\nu) \end{align*} whenever $\epsilon < 2^{-10({\mathbf D}(\mu\,\|\,\nu)+1)}$.

In words, if $\mu$ is not too divergent from $\nu$ then diluting $\nu$ a little should not decrease the divergence more than a constant factor.

Please let me know if this has been studied somewhere, has a name, or related to well known principles, or false.

Thanks