Sharp Poincaré–Wirtinger inequalities on complete graphs

Abstract

Let \(K_n=(V,E)\) be the complete graph with \(n\geq 3\) vertices (here \(V\) and \(E\) denote the set of vertices and edges of \(K_n\) respectively). We find the optimal value \({\bf{C}}_{n,p}\) such that the inequality
\(\|f-m_f\|_p\le {\bf C}_{n,p}\operatorname{var}_{p}f\)

holds for every \(f\colon V\to \mathbb{R},\) where \(\operatorname{var}_p\) stands for the \(p\)-variation, and \(m_f\) stands for the average value of \(f\), for all \(p\in[1,3+\delta^1_n)\cup (3+\delta^2_n,+\infty)\), for \(\delta^1_n=\frac{1}{2n^2\log(n)}+O(1/n^3)\) and \(\delta^2_n=\frac{2}{n}+O(1/n^2)\). Moreover, we characterize all the maximizer functions in that case. The behavior of the maximizers is different in each of the intervals \((1,2)\), \((2,3+\delta^{1}_n)\) and \((3+\delta^{2}_n,\infty)\).