For a probability measure, convexity puts the function of the mean below the mean of the function.Suggest a slogan
Let \((X,\mcm,\mu)\) be a measure space with \(\mu(X)=1\), let \(-\infty\leq a<b\leq\infty\), and let \(\phi\colon(a,b)\to\RR\) be convex, that is, \[ \phi(tx + (1-t)y) \leq t\phi(x) + (1-t)\phi(y) \quad\text{for all } x,y\in(a,b) \text{ and } t\in[0,1]. \] If \(f\in L^1(\mu)\) is real-valued with \(a<f(x)<b\) for all \(x\in X\), then \[ \phi\Big(\int_X f\dmu\Big)\leq\int_X\phi\circ f\dmu . \]