Equality conditions for correlation inequalities

We prove equality conditions for the Ahlswede--Daykin (AD) inequality and the Fortuin--Kasteleyn--Ginibre (FKG) inequality. We then present a number of applications and special cases of these equality conditions. These include Bj\"orner's and Fishburn's inequalities for linear extensions of finite posets, the Lam--Postnikov--Pylyavskyy (LPP) and the Okounkov inequalities for Schur positivity of products of Schur functions. We conclude with equality conditions for the Ahlswede--Daykin--Schur (ADS) inequality recently introduced in Chan--Chen--Pak--Soskin (2026), which is an AD type extension of the LPP inequality.