Variance inequalities using first derivatives

We develop variance inequalities on functions of random variables using mild information such as the first derivatives. Specifically, when ff and gg are absolutely continuous functions, we show that Var[f(X)]≤Var[g(X)]Var[f(X)]≤Var[g(X)] for any random variable XX if and only if ff is a function of gg and |f′|≤|g′||f′|≤|g′| almost everywhere. Our results provide a form of master inequality from which other variance inequalities can be obtained.