# Trying to prove an inequality (looks similar to entropy)

I'm trying to prove the following inequality (or something similar, up to a constant factor in either side of the inequality): $$k\cdot\sum_{i=1}^{k}x_{i}\cdot\ln\left(x_{i}\right)\geq\sum_{i=1}^{k}x_{i}\cdot\left(x_{i}-1\right)$$ where $$\forall i\in\left[k\right]$$, $$x_i \in\left[0,k\right]$$ (the $$x_i$$s are not necessarily natural numbers, but we can assume that they're rational if it helps), and $$\sum_{i=0}^k x_i=k$$.

I've tried plotting it for $$k=2,3$$ and ran some numerical experiments for larger $$k$$, and I'm 99% sure this inequality is correct, but I'm still struggling with the proof.

Up to some normalizing, I find the left-hand side quite similar to the entropy of a probability distribution, but I didn't manage to take advantage of this fact either. I also tried looking for inequalities that only hold on simplex-like hyperplanes, but couldn't find anything useful.

Any ideas? Thanks!

The function $$f(x) = k x \ln(x) - x(x-1)$$ has the second derivative $$f''(x) = \frac k x - 2$$ so that it is convex on the interval $$[0, k/2]$$. If all $$x_i$$ are in the interval $$[0, k/2]$$ then Jensen's inequality can be applied, so that $$\sum_{i=1}^k f(x_i) \ge k f\left( \frac 1k \sum_{i=1}^k x_i\right) = k f(1) = 0 \, ,$$ which is the desired estimate.
It remains to investigate the case where $$x_i > k/2$$ for one $$i$$, say $$x_k > k/2$$. Then $$\sum_{i=1}^k f(x_i) = \sum_{i=1}^{k-1} f(x_i) + f(x_k) \ge (k-1) f\left( \frac 1{k-1} \sum_{i=1}^{k-1} x_i\right) + f(x_k) \\ = (k-1) f\left( \frac {k-x_k}{k-1} \right) + f(x_k) \, .$$ Therefore we define $$g(x) = (k-1) f\left( \frac {k-x}{k-1} \right) + f(x) \, .$$ An elementary calculation shows that $$g(1) =g'(1) = 0$$, and $$g''(x) = \frac{2k(x^2-kx+\frac{k^2-k}{2})}{(k-1)x(k-x)} \ge 0 \, .$$ It follows that $$g(x) \ge 0$$ on $$[0, k]$$, and that completes the proof.
• This only works for $x_i \geq 1$, otherwise we have $x_i - 1<0$, and the rightmost inequality in your answer doesn't hold. Sep 2, 2021 at 9:29