A discrete Poincare inequality Given a (bounded) sequence $\{q_n\}_{n\geq 0}$ such that $|q_n| \leq 1$ for all $n \geq 0$ and $\sum_{n\geq 0} q_n = 0$. We can impose the condition that $\sum_{n\geq 0} |q_n| \leq 2$ as well. I am wondering whether there exists a fixed constant, independent of the sequence $\{q_n\}$, such that $$\sum_{n \geq 0} |q_n - q_{n+1}|^2 \geq C \sum_{n \geq 0} |q_n|^2$$ holds true?

Edit: It seems that $\sum_{n\geq 0} |q_n| \leq 2$ might not be enough, I am wondering if higher order moment bound such as $\sum_{n\geq 0} (1+n)|q_n| < \infty$ will suffice.
 A: Fix $n \in \mathbb{N}$. Define the vector $\mathbf{q} = (q_1, q_2, \dots q_n) \in \mathbb{R}^n$ to be
$$
q_i = \cos \left( \frac{\pi}{n} (i - 1/2) \right)
$$
One can check that this vector has the properties that

*

*$\sum_{i=1}^n q_i = 0$

*$\sum_{i=1}^n (q_{i} - q_{i+1})^2 = C_n \cdot \sum_{i=1}^n q_i^2$
where $C_n = 2 (1 - \cos(\pi / n))$, which goes to $0$ as $n$ increases.
Moreover, observe that these properties are invariant to scaling the vector, i.e. $\alpha \cdot \mathbf{q}$ for some $\alpha \in \mathbb{R}$ so that $\alpha \neq 0$.
On the other hand, the conditions you are considering (i.e. originally $\sum_{i=1}^n (1+i) |q_i| < \infty$, and in the updated question $\sum_{i=1}^n |q_i| \leq 2$) do depend on the scaling.
So, you can just scale $\mathbf{q}$ appropriately to satisfy either of these two conditions and it will still be a "counterexample" in the sense that it will satisfy conditions (1) and (2) above.
So, it seems that the statement you are looking to prove is not true.
However, I'm not very familiar with Poincare inequalities so it could be that there is a different analogous statement that you may be interested in.

Here is how I constructed this example. First, observe that you are asking for a lower bound on the Rayleigh quotient
$$\frac{ \mathbf{q}^\top \mathbf{A} \mathbf{q} }{ \mathbf{q}^\top \mathbf{q} } = \frac{\sum_{i=1}^n (q_{i} - q_{i+1})^2}{\sum_{i=1}^n q_i} ,$$
where $\mathbf{A}$ is the Laplacian matrix of the $n$-length path graph.
Its smallest eigenvalue is $0$ which corresponds to an eigenvector which is the constant vector.
You specify that we are interested in $\mathbf{q} \in \mathbb{R}^n$ satisfying $\sum_{i=1}^n q_i = 0$, which is orthgonal to the constant vector.
By Courant-Fischer theorem, I can minimize the Rayleigh quotient (subject to $\sum_{i=1}^n q_i = 0$)  by taking $\mathbf{q}$ to be the second smallest eigenvector. The minimal value of the Rayleigh quotient will be the second smallest eigenvalue of $\mathbf{A}$.
Because we know the eigenvalues and eigenvectors of the Laplacian matrix of the path graph, we can get the exact answer.
