Unknown

220 7. Approximations and Inequalities

(e) the Bernstein polynomial of degree 2 (modified to the interval)

(f) the Taylor approximants of degrees 1, 2 and 3 at 100, 121, 144.

In particular, use the above approximants to estimate (125)l/‘, and com-

pare your results with the true answer.

7.3 Inequalities

Often, an important consideration in comparing two functions is to deter-

mine when one exceeds the other. One of the most useful inequalities is

that between the arithmetic and geometric means of positive quantities.

For nonnegative reals al, a~,... , a,, the geometric mean is defined to be

(Ql(132.. .a,) l/n

and the arithmetic mean to be

(al + a2 +... + a,)/n.

The Arithmetic-Geometric Means (AGM) Inequality asserts that

(ala2... ata) 1’n < (a1 + a2 +. -. + a,)/n

with equality if and only if all the oi are equal. This has already been
established in the cases n = 2 and n = 3 (Exercises 1.2.17 and 1.5.9)
by showing that x2 + y2 - 2xy 2 0 and x3 + y3 + .z3 - 32~~ 1 0 when
x, y, z 2 0. For inequalities in general, one useful strategy is to write an
appropriate function as a sum or product of polynomials known to be
positive, such as squares of other polynomials. This was also the basis of
one of the arguments (Exercise 1.2.15) used in establishing the Cauchy-
Schwarz-Bunjakovsky (CSB) Inequality:

with inequality iff the ratio of the ui is equal to the ratio of the bi.
In the exercises, we will establish some other useful inequalities.

Exercises

1. (a) Write z4 + y4 + .r4 + w4 - 4xyzw in the form up2 + bq2 + cr2
   where a, b, c are positive constants and p, q, T are polynomials
   over R in x, y, z, w.
   (b) Establish the AGM inequality for n = 4.