RI Chapter 7 Inequalities

RAFFLES INSTITUTION H3 Mathematics 9820 _____________________________________________________________________________________________ _________________________ Chapter 7: Inequalities Page 1 of 18 Chapter 7 Inequalities The study of inequalities is important in mathematics. For starters, we often need to compare the magnitude of two mathematical objects. This is essentially with the aid of inequalities. You will find that in university, the use of inequalities in analysis is particularly important. For example, in describing the concept of a “limit” and “convergence”. SYLLABUS INCLUDES  Equations and inequalities (such as Triangle inequality, AM-GM inequality, Cauchy- Schwarz Inequality) CONTENT 1 Introduction 1.1 Estimation 1.2 Absolute values, estimating size 2 Triangle Inequality 3 Arithmetic-Mean and Geometric Mean (AM-GM) inequality 4 Cauchy-Schwarz Inequality 5 Inequalities involving other functions 5.1 Inequalities involving monotonic functions 5.2 Inequalities involving convex functions

Raffles Institution H3 Mathematics _______________________________________________________________________________________________________ _______________________ Chapter 7: Inequalities Page 2 of 18 1 Introduction In general, inequalities are used to make comparisons, and to make estimations. It is one of two simple tools that form the language of estimations, the other being absolute values for measuring size and distance. 1.1 Estimations One of the major uses of inequalities, as we mentioned, is to make estimations. These have a language of their own that you need to get used to. Definition 1.1.1 If c is a number we are estimating, and m < c < M, we say that m is a lower estimate (lower bound) for c and M is an upper estimate (upper bound) for c. If two sets of upper and lower bounds satisfy the inequalities ''m m c M M    we say that ', 'mM are stronger or sharper estimates for c, while m, M are weaker estimates. Estimates are obtained in many ways; the inequality laws that you have been introduced to in H2 Mathematics usually play an important role. Calculus can also be used. Here are some examples to illustrate. Example 1 Give upper and lower bounds for 42 1 31aa . Solution We are not told what a is, so our bounds will have to be valid no matter what a is, i.e. valid for all a. Let us start with the denominator. Since any square is nonnegative, the polynomial has its smallest value 1 when a = 0, and it has no upper bound since a can be arbitrarily large. Hence we have 421 3 1aa   which implies 42 101 31aa  These are the sharpest possible estimates valid for all a: In fact, the upper bound 1 is attained when a = 0, and the lower bound is 0 since the fraction can be made arbitrarily close to 0 by taking a sufficiently large. Exercise 2 Give upper and lower estimates for 2 2 1 sin 1 cos n n   , for 0n .

Raffles Institution H3 Mathematics _______________________________________________________________________________________________________ _______________________ Chapter 7: Inequalities Page 3 of 18 Exercise 3 By interpreting the integral as the area under the graph of 1 xy over the interval [1, 2], estimate 2 1 dln 2 . x x 1.2 Absolute values. Estimating size. We now look to the other tool of estimation, the absolute value. As with inequalities, we have introduced the basic rules in manipulating them in H2 Mathematics. Here are two good ways to think about absolute value. Absolute value measures magnitude: |a| is the size of a. Negative numbers can be big too: a million dollar loss is a big sum for a small business. Absolute value measures distance: |a – b| is the distance between a and b. The latter is often used to describe intervals on the real axis. For instance, the interval (2, 4) can be described as {x: |x – 3| < 1}, i.e. the set of points whose distance from the point 3 is at most 1. Two important and common use of the absolute value are as follow: (1) 0a  for all real values a. (2) a M M a M    . The absolute value is also an efficient way to give symmetric bounds. In fact, the inequality on the left is often more convenient to use. In the bounds are not symmetric to start with, they can be made so by doing the following:  , where max ,K a L a M M K L     . In working with absolute values, we will make frequent use of the simple property |ab| = |a||b| as well as the triangle inequality, a law that connects the absolute value with sums.

Raffles Institution H3 Mathematics _______________________________________________________________________________________________________ _______________________ Chapter 7: Inequalities Page 4 of 18 2 Triangle Inequality Theorem 2.1 (Triangle Inequality) For real numbers a and b, with equality if and only if 0.a b a b ab    Proof Other forms of Triangle Inequality Exercise 4 Show that (a) (hint: )a b a b a a b b      (b) a b a b   Exercise 5 If 3 and 1ab what is a lower “estimate” for ab and ab ? Exercise 6 In Fourier Analysis, one uses trigonometric sums of the form 12cos cos 2 ... cosnnS c t c t c nt    If 1 2 i ic  , give an upper bound for Sn.

Raffles Institution H3 Mathematics _______________________________________________________________________________________________________ _______________________ Chapter 7: Inequalities Page 5 of 18 The triangle inequality, as the name suggests, has a geometric interpretation. It states that the sum of the lengths of any two sides of a triangle is greater than the length of the remaining side. We have seen the proof of the triangle inequality in the case of real numbers above. If we extend it to a higher dimension, and replace the variables with vectors and the absolute value with the norm (distance function), we have   a b a b , where the length of the third side is represented by ab . The proof, however, is best done in the following manner:       22 22 2 2 2 2 2 22 cos a b a b a b a b a b a b a a b b a a b b a a b b a b a b a b a b                       which is true since cos 1  . There is of course a geometrical proof, which you can read up easily if you are interested. The triangle inequality can be extended by mathematical induction to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints. Consequently, the length of any polygon side i s always less than the sum of the other polygon side lengths, i.e. 1 2 1 2 ... ...a a a a a a nn       . Exercise 7 Determine all triangles whose side lengths are in (i) arithmetic progression, (ii) geometric progression.

Raffles Institution H3 Mathematics _______________________________________________________________________________________________________ _______________________ Chapter 7: Inequalities Page 6 of 18 For the remaining sections of this chapter, we will look at two classical inequalities as well as a couple of other methods that can be used to provide estimates. Let us first begin with the AM- GM inequality. 3 Arithmetic Mean-Geometric Mean (AM-GM) Inequality The arithmetic mean (AM) of n real numbers is defined as the average of the n numbers. In the context of this section, we will consider the case where the numbers are nonnegative. Thus for 12 12 ..., ,..., 0, then AM n n x x xx x x n    . The geometric mean (GM) of the n real numbers is like the average of the product, thus for   1 1 2 1 2, ,..., 0, then GM ... n nnx x x x x x  . In fact, the arithmetic and geometric means are special cases of the r-th power means, rP , which are