ABSTRACT
This chapter contains an account of some monotonicity and convexity theorems within the context of the first and second symmetric derivatives. It discusses the basic monotonicity theorem for the symmetric derivative in a number of variants and a number of convexity theorems. Convexity theorems are a natural companion for monotonicity theorems in general; their discussion is particularly natural since they can be obtained from assertions about the second symmetric derivative just as our monotonicity theorems derive from assertions about the first symmetric derivative. As an application of the Freiling monotonicity Theorem presents an independent proof. The basic monotonicity theorem to be expected is that a measurable function with a nonnegative approximate symmetric derivative is equivalent to a nondecreasing function.
