Why study inequalities? Richard Bellman [Bellman, 1978] answered:

There are three reasons for the study of inequalities: practical, theoretical, and aesthetic.

In many practical investigations, it is necessary to bound one quantity by another. The classical inequalities are very useful for this purpose.

From the theoretical point of view, very simple questions give rise to entire theories. For example, we may ask when the nonnegativity of one quantity implies that of another. This simple question leads to the theory of positive operators and the theory of differential inequalities. The theory of quasilinearization is a blend of the theory of dynamic...

A simple, common, but very powerful tool for illustrating inequalities among positive numbers is to represent such numbers by means of line segments whose lengths are the given positive numbers. In this chapter, and in the ones to follow, we illustrate inequalities by comparing the lengths of the segments using one or more of the following methods:

1.The inclusion principle.Show that one segment is a subset of another. We will generalize this method in the next chapter when we represent numbers by areas and volumes and illustrate inequalities via the subset relation.

2.The geodesic principle.Use the...

We now extend the inclusion principle from the first chapter by representing a positive number as the area or the volume of an object, and showing that one object is included in the other. We begin with area representations of numbers, primarily by rectangles and triangles.

The inclusion principle will be applied in two ways. Suppose we wish to establish$x\underline < y$and we have found a regionAwith areaxand a regionBwith areay. Then$x\underline < y$if either (a)Afits insideB, or (b) pieces ofBcoverAwith possible overlap of some pieces....

Since the time of Euclid, geometers have studied procedures for constructing triangles given elements such as the three sides, two sides and an angle, and so on. The construction procedure usually has a constraint, such as using only an unmarked straightedge and compass to draw the triangle. There are constraints on the given elements as well, usually given as inequalities. For example, as noted in Section 1.1, a triangle with sides of lengtha;b, andccan be constructed if and only if the three triangle inequalities$a < b + c,$$b < c + a,$and$c < a + b$hold.

In this chapter we examine inequalities among...

For many geometric inequalities, the strategy of inscribing or circumscribing a figure can be useful (recall, for example, Application 1.1, Sections 1.3 and 1.6, and Theorem 2.3). Of all such inscribed or circumscribed figures, the circle plays a central role, and results in a variety of inequalities relating the radius of the circle to numbers associated with the given figure, such as side lengths, perimeter, area, etc.

The triangle is exceptional because every triangle possesses a circle passing through the vertices of the triangle, thecircumcircle, whose center is thecircumcenterof the triangle, and a circle inside the triangle...

Anisometryin the plane is a transformation that preserves distances. The primary isometries are reflections, rotations, and translations. Preservation of distance implies that isometries preserve angles and areas, so shapes are invariant. As we shall see in this chapter and the next, using isometries is a powerful method for proving geometrical inequalities. Reflections will be used for three different purposes, which we illustrate in the next three examples.

Example 5.1. Minimal paths

Given two pointsAandBon the same side of a lineL, what pointCinLmakes$\left| {AC} \right| + \left| {CB} \right|$a minimum?

UsingLas a...

Like reflections, rotations are useful isometries, since in addition to preserving distances and angles they preserve orientation. As we shall see in this chapter, rotating figures is a useful way to create a visual explanation of an inequality.

In rotation, a figure or portion of a figure is rotated in the plane about a given point through a given angle, as illustrated in the following example.

Example 6.1.A maximal distance problem

LetABCbe an equilateral triangle whose sides have lengtha. Suppose a pointPis located a fixed distanceufrom vertexA, and a fixed distance...

In the two previous chapters, we examined how reflections and rotations (isometric transformations) may be used in a visual approach to inequalities. Non-isometric transformations—transformations that do not necessarily preserve lengths—constitute an interesting class of mappings for proving some inequalities. We consider three types of non-isometric transformations in this chapter: similarity of figures, measure-preserving transformations, and projections. The first preserves shapes, but changes measures by a given factor or its powers while the others may change shapes of figures but preserve other properties.

In 1935, the following problem proposal appeared in the Advanced Problems section of theAmerican Mathematical...

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 amoving frameto 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.

LetSandTbe...

In this final chapter, we examine some methods for illustrating inequalities by combining two or more techniques from earlier chapters. We also give a brief introduction to the theory of majorization, which has proven to be remarkably effective in proving inequalities.

We have seen in previous chapters several examples where a final inequality is obtained after illustrating several simpler inequalities and then combining them. See, for example, Sections 4.1 and 4.3. We now revisit this methodology.

Example 9.1.If a, b, c are the sides of a right triangle with hypotenuse c, then$c\left( {a + b} \right) > 2\sqrt {2ab} $.

Consider the square in Figure 9.1....