Many simple properties of real-valued functions, such as boundedness, monotonicity, convexity, and the Lipschitz condition, can be expressed in terms of inequalities. Consequently there are visual representations of many of them, some of which are familiar. In this chapter we introduce the idea of a moving frame to illustrate some of these properties, and then use them to establish additional inequalities. We also investigate the role played by the convexity or concavity of a function in establishing functional inequalities. We conclude the chapter by examining inequalities in which areas under graphs of functions represent numbers.

Boundedness and monotonicity

Let S and T be subsets of the reals. A function f : S → T is bounded if there exist constants m and M such that for all x in S, m ≤ f(x) ≤ M. Visually, this means that the entire graph of y = f(x) lies between the horizontal lines y = m and y = M, as illustrated in Figure 8.1a. A moving frame is like a window, in this case with height M - m units and some convenient width, such that, as the frame is moved horizontally with the opening always between the lines y = m and y = M, we see the graph inside the frame, never in the opaque regions shaded gray in Figure 8.1b.