Trying to prove an inequality (looks similar to entropy)

Question

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!

Answer 1

Expanding a little bit on fedja's comment, it is actually convenient to consider the inequality in the equivalent form $$ \sum_{i=1}^k ( x_i \ln x_i - x_i + 1) \geq c_k \sum_{i=1}^k (x_i-1)^2 , $$ where we are interested to find $c_k$ such that the inequalities hold for all $x_i \in [0,k]$ with $\sum_{i=1}^k x_i = k$.

By Taylor, it is easy to see, that we have the two inequalities $$ x \ln x -x +1 \geq \tfrac{1}{2} (x-1)^2 \qquad\text{ for } x \in [0,1] $$ and $$ x \ln x -x +1 \leq \tfrac{1}{2} (x-1)^2 \qquad\text{ for } x \geq 1 $$ Hence, we are only in trouble for $x\in [1,k]$. By comparing the derivatives of the function on the left hand and right hand side: $\ln(x)$ and $x-1$, we also see that the defect in the second estimate is monotone increasing in $x$ starting at $x=1$, where both are equal.
Hence, we find the optimal constant as $$ c_k = \inf_{x\in [0,k]} \frac{x \ln x -x +1}{(x-1)^2} = \frac{k \ln k - k +1}{(k-1)^2}. $$ Note that $c_k$ is monotone decreasing in $k$ with $c_1=1/2$. In particular, it holds $c_k \geq \frac{1}{2k}$, but this gives not the correct asymptotics, since for $k\geq 2$, one can use the lower bound $$ c_k \geq \frac{\ln k -1}{k} , $$ which is asymptotic sharp for $k\to \infty$.