TITLE: Entropy Lower Bound in Terms of $\ell_2$ norm
QUESTION [6 upvotes]: Define
$$
\begin{align}
H(p_1, \dots, p_n) &= \sum_{i=1}^n p_i\log1/p_i\\
&=\log n+\sum_{i=1}^n\sum_{k=2}^\infty (-1)^{k + 1} n^{k - 1} \frac{(p_i - 1/n)^k}{k (k - 1)},
\end{align}
$$
where $p_1,\dots,p_n\ge0$ sum to $1$.
Then we have the classic inequality
$H(p_1,p_2)\ge(\log2)(1-2((p_1-1/2)^2+2(p_2-1/2)^2))=(\log 2)(1-2\|p-1/2\|^2)$, and we might wonder if that could be extended for $n>2$.
In particular with something like
$$\begin{align}
H(p_1,\dots,p_n)&\ge(\log n)(1-c_n\|p-1/n\|^2_2).
\end{align}$$
From experiments with $n=3$, it seems like
$c_n\ge\frac{2 n (\log n/2)}{(n-2) \log n}=2(1-O(1/\log n))$ suffices, but I don't have a proof of this. It is also slightly inconvenient that it can go below $0$, something that wasn't the case with the $n=2$ case.
Bounding the terms individually, we can get $H(p_1,\dots,p_n)\ge-2+4\sum_{i=1}^n\frac{p_i}{1+p_i}$, which is non-negative, but not as relatable to the $\ell_2$ norm. We can also bound $H\ge n/4-\|p-1/2\|_2^2$, but somehow bounding centered in $1/n$ seems more natural.
Is there a well known lower bound like this, relating $H(p)$ with $\|p\|_2$? Ideally, one that is asymptotically tight at $p_1=\dots=p_n=1/n$ and is always positive.

REPLY [3 votes]: This is just scribbling down some observations in the comments, to have them more, well, visible.

As observed by K B Dave, the inequality of leonbloy is an instance of $\mathrm{KL} \le \chi^2.$ This can thus be improved by using a tighter inequality of this form. Note that the resulting inequality was mentioned by Clement in the original comments.
We have $$\mathrm{KL}(P\|Q) = \int \log \frac{\mathrm{d}P}{\mathrm{d}Q} \,\mathrm{d}P \le \log \int \left( \frac{\mathrm{d}P}{\mathrm{d}Q}\right) \,\mathrm{d}P = \log \int \left( \frac{\mathrm{d}P}{\mathrm{d}Q}\right)^2 \,\mathrm{d}Q = \log (1 + \chi^2(P\|Q)), $$ where the inequality follows by the concavity of $\log$ and Jensen's inequality. Applying the above to this case yields $$H(P) = \log n - \mathrm{KL}(P\|\mathbf{1}/n) \ge \log n - \log(1 + \sum_x  \frac{(P_x - 1/n)^2}{1/n}) = \log n - \log (1 +  n\|P - \mathbf{1}/n\|_2^2).$$
Note that if $n\|P - \mathbf{1}/n\|_2^2 \ll 1,$ we may use $\log (1 + x) \approx x$ for small values of $x$ to recover the original bound.
We may also state the above bound in the following equivalent way: Observe that $1 + \chi^2(P\|Q) = \mathbb{E}_Q[ ({\mathrm{d}P}/{\mathrm{d}Q})^2].$ Again plugging in $Q$ uniform on $n$ letters yields that the latter expectation is simply $n\|P\|_2^2,$ and we get that 
$$ H(P) \ge - \log \|P\|_2^2,$$ the form stated by Clement. Note that $\|P\|_2 \le 1$ for every distribution, so the form above makes it obvious that the inequalities are non-trivial (in that the RHS is $\ge$ 0) for every $P$.