A Triangle Inequality for Covariances of Binary FKG Random Variables

For binary random variables $\sigma_1, \sigma_2, \ldots, \sigma_n$ that satisfy the well-known FKG condition, we show that the variances and covariances satisfy $\operatorname{Var}(\sigma_j) \operatorname{Cov}(\sigma_i, \sigma_k) \geq \operatorname{Cov}(\sigma_i, \sigma_j)\operatorname{Cov}(\sigma_j, \sigma_k),\quad 1 \leq i, j, k \leq n.$ This generalizes and improves a result by Graham for ferromagnetic Ising models with nonnegative external fields.