Let $X$ be a random variable with probability mass function $p(x) = \mathbb{P}[X = x]$. I know entropy $H(X)$ of $X$ measures the uncertainty of $X$ and a large value of $H(X)$ means $p(x)$ is nearly uniform.

To me, it seems to be likely that $D(X) := \max_x p(x) - \min_x p(x)$ can be another measure of uncertainty; small $D(X)$ corresponds to $p(x)$ close to a uniform distribution.

So here is my question. Is there any connection (e.g. inequality) between $H(X)$ and $D(X)$?

I've tried this and that, and all I've got so far is the following:

For simplicity, let $p(x)$ be a Bernoulli distribution $\mathrm{Bin}(1, \theta)\ (\theta \in [0, 1])$: $$p(x) = \begin{cases} \hfill\theta &(x = 1) \\ \hfill1 - \theta &(x = 0)\end{cases}$$ and $\tilde{p}(x)$ be a "flipped" version of $p(x)$: $$\tilde{p}(x) = \begin{cases} \hfill1 - \theta &(x = 1) \\ \hfill\theta &(x = 0).\end{cases}$$

$D(X)$ is related to the $L^1$ (or total variation) divergence between $p$ and $\tilde{p}$ as follows: $$ \lVert p - \tilde{p} \rVert_1 = \lvert \theta - (1 - \theta) \rvert + \lvert (1 - \theta) - \theta \rvert = 2 D(X) . $$ From Pinsker's inequality, we have \begin{align*} D(X)^2 = \, & \frac{1}{4} \lVert p - \tilde{p} \rVert_1^2 \\ \leq \, & \frac{1}{2} \mathrm{KL}(p||\tilde{p}) \\ = \, & \frac{1}{2} \left( \theta \log \left( \frac{\theta}{1 - \theta} \right) + (1 - \theta) \log \left( \frac{1 - \theta}{\theta} \right) \right) \\ = \, & \frac{1}{2} \left\{ \left( \theta \log \theta + (1 - \theta) \log (1 - \theta) \right) - \left( \theta \log (1 - \theta) + (1 - \theta) \log \theta \right)\right\} \\ = \, & \frac{1}{2} \left\{ \left( \theta \log \theta + (1 - \theta) \log (1 - \theta) \right) + \left( (1 - \theta) \log (1 - \theta) + \theta \log \theta \right) - \left( \log (1 - \theta) + \log \theta \right) \right\} \\ = \, & \frac{1}{2} \left\{ -2 H(X) - \left( \log (1 - \theta) + \log \theta \right) \right\} \\ = \, & - H(X) - \frac{1}{2} \log \theta(1 - \theta). \end{align*}

This indeed represents the relationship between $H(X)$ and $D(X)$ (i.e. large $H(X)$ = small $D(X)$). However, it has several limitations:

- It is limited to cases where $X$ is a binary variable.
- I don't like the extra term $\log \theta(1 - \theta)$.

Does anyone have a more general & interpretable relationship between $H(X)$ and $D(X)$ (or some other measures similar to $D(X)$)?