## ABSTRACT

In the case of many important problems for partial differential equations which arise in mathematical physics, it is sometimes possible to define an integral measure (norm) of the solution, which is found to satisfy a second order differential inequality, together with appropriate side conditions. This chapter begins with a brief survey of simple convexity and concavity of a function – corresponding to the simplest second order differential inequality. It discusses logarithmically convex functions, that is, ones with convex logarithms and functions which satisfy differential inequalities similar to those satisfied by such functions. The chapter considers a particularly useful cross-sectional measure for two-dimensional regions; it arises in a number of contexts and is particularly appropriate to regions with convex boundaries. Convexity and concavity are complementary, mirror-image concepts, which are defined by the simplest second order differential inquality.