# Lower bounding mutual information by entropy rate

I encounter the following problem in my research and I hope you would kindly help me with it. Here is the description:

Let $$M$$ be an $$m \times n$$ stochastic matrix with each row summing up to $$1$$. An important feature of $$M$$ is that the rows $$\{\mathbf{m}_i^{T}\}_{i=1,\dots, n}$$ of $$M$$, which are probability vectors of dimension $$n$$, are all distinct. Now let $$\mathbf{x}$$ be a probability vector of dimension $$m$$, and let function $$f$$ be defined as: $$f(a) = \min_{h(\mathbf{x})=a}h\left(\mathbf{x}^{T}M\right) - \sum_{i=1}^{m}x_ih(\mathbf{m}_i),$$ where $$h$$ is the entropy that is $$h(\mathbf{x}) = -\sum_{i=1}^{m}x_i\log x_i$$where $$\mathbf{x}=(x_1,\dots, x_m)$$.

We note that, denoting $$\mathbf{x}^{T}M$$ by $$\mathbf{y}$$, we have in fact $$f(a) = \min_{h(\mathbf{x}) = a}I(\mathbf{x};\mathbf{y}).$$

Here are the properties I want to know about $$f$$:

1. Is $$f$$ continuous and differentiable?
2. By concavity of entropy, we have $$f \geq 0$$. However, if we have $$\mathbf{m}_i = \mathbf{m}_j$$, we do have $$f(a)=0$$ while $$h(a)\neq 0$$. For example let $$\mathbf{m}_1 = \mathbf{m}_2$$, then $$\mathbf{x} = \left(.2, .8, 0,\dots, 0\right)$$, or any probability vector with all entries $$x_3,\dots, x_m=0$$, makes $$f(h(\mathbf{x}))=0$$. But what happens if all $$\mathbf{m}_i$$s are distinct? To be more specific, if $$f$$ is differentiable, what is $$f'(0)$$? Or do we have something like $$f(a)\geq c \cdot h(a),$$ where $$c>0$$ is a constant?