Circular Rearrangement Inequality and Optimal Cyclic Birth and Death Chains

We prove a generalized circular rearrangement inequality: among all circular arrangements of a finite collection of positive numbers, the greedy arrangement simultaneously maximizes the sums of products of $k$ consecutive entries for every $k$. This resolves a 2023 conjecture of Holmes-Holroyd-Ramírez and extends the classical circular rearrangement inequality for products of adjacent entries. As an application, we show that the greedy ordering minimizes the speed of a cyclic birth-and-death chain, resolving another conjecture of the same authors.