Given two distributions $x=(x_1,\ldots, x_n),y=(y_1,\ldots y_n)$ on $[n]$, it is known by Fannes inequality that $H(x)-H(y)\leq O(\|x-y\|_1\log n)$, where $H(\cdot)$ and $\|\cdot\|_1$ represent Shannon entropy and $\ell_1$ distance, respectively. I wonder under what conditions we can replace the upper bound by a dimension-independent one, like $\|x-y\|_1^c$. A concrete example I am interested is the following.
Let $\sigma,\tau$ be two permutations on $[n]$. Define $\sigma(x)=(x_{\sigma(1)},\ldots, x_{\sigma(n)})$. Can we show that $$H(\frac{\sigma(x)+\tau(x)}{2})-H(x)\leq O(\|\frac{\sigma(x)+\tau(x)}{2}-x\|_1^c)?$$
In the special case that $\tau=id$ the inequality holds. Because in this case $$LHS=JS(\sigma(x),x)\leq \|x-\sigma(x)\|_1,$$ where $JS()$ is the Jensen diverngence.