Li and Yorke introduced in their fundamental paper the term chaotic for a class of self-mappings of an interval. Following the Li-Yorke result that period three implies chaos, many authors worked on periodic conditions that allow the same conclusion. Li and Yorke also introduced four-point inequalities satisfied by a point and its three successors with respect to the given function f. They showed that these imply the existence of a three-period and hence chaos. The authors of this article began investigations under the US Army Summer Faculty Research and Engineering Program 1983, and showed that the Li-Yorke inequalities play a fundamental role in the theory of chaos. For this article they have singled out three theorems. The first is an addendum to the Li-Yorke theorem. The second establishes equivalent companion inequalities to the Li-Yorke inequalities. The third theorem, of a different character, is especially important in applications. We also introduce the elementary notion of a Newton function our term which we have found indispensable in the investigation of parameter families and which, we think, deserves to be better known.