Question 34
Let $n$ be a given integer which is greater than $1$. Find the greatest constant $\lambda(n)$ such that for any non-zero complex $z_1, z_2, \dots, z_n$, we have
$$
\sum_{k=1}^{n} |z_k|^2 \geq \lambda(n) \min_{1 \leq k \leq n} \{|z_{k+1} - z_k|^2\},
$$
where $z_{n+1} = z_1$.
$$
\sum_{k=1}^{n} |z_k|^2 \geq \lambda(n) \min_{1 \leq k \leq n} \{|z_{k+1} - z_k|^2\},
$$
where $z_{n+1} = z_1$.