I wonder if people have studied the characterization of data processing inequality. Here's the data processing inequality for mutual information: 3 variables for Markov chain: $X\to Y\to Z$, then $I(X, Y)\geq I(X, Z)$.
What are the other functions that have this property? I'm looking for characterization of function $f$ such that $f(X, Y)\geq f(X, Z)$.