Functional inequality

Summary

A functional inequality is an inequality between expressions in an unknown function. It is required to hold for all values of the variables. It is the inequality analogue of a functional equation. Standard examples are subadditivity $f(x + y) \le f(x) + f(y)$ and midpoint convexity $f\big(\tfrac{x + y}{2}\big) \le \tfrac{f(x) + f(y)}{2}$. Typical questions are which functions satisfy a given inequality, and when mild regularity forces its solutions to be nice.