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Unformatted text preview: l p, q where p and q.
Define
hy hI
A
yq
yI
and
hy hI
py .
B
yI
Since h is a convex function we know that whenever p y I z q p y z I we have
hy
y hI
I 284 hz
z hI
I q and therefore sup A
properties. inf B and any number k in the interval sup A, inf B will have the desired b. If k is chosen with the property specified in part a then for every number x
0, 1 we have
hfx
hI
kfx I.
The desired inequality becomes clear when we consider the cases f x I, f x I and
f x I.
c. Integrating both sides of the preceding inequality yields the inequality Þ0 h f x Þ 0 f x dx 1 1 dx h . This inequality is known as Jensen’s inequality.
When we integrate both sides we obtain Þ0 kÞ 1 hfx hI dx 1 fx I dx 0 which yields Þ0 h f x
1 dx hI 0 and this is the desired result.
6. a. Prove that if f is integrable on the interval 0, 1 then Þ 0 exp f x
1 Þ 0 f x dx
1 dx exp . This inequality follows at once from Jensen’s inequality because the function exp, having an
increasing derivative, must be convex.
b. Prove that if f is integrable on the interval 0, 1 and if for some number 0 we have f x for every
x
0, 1 then Þ 0 f x dx
1 Þ 0 log f x
1 exp dx . Since the function log is uniformly continuous on the interval , Ý we know that the function
g log f
is integrable on 0, 1 . It follows from Part a that Þ 0 exp g x dx exp Þ 0 f x dx exp Þ 0 log f x 1 Þ 0 g x dx
1 which gives us
1 1 dx . c. Given positive numbers c 1 , c 2 , , c n , apply part b to an appropriate step function f on 0, 1 to obtain the
inequality
c1 c2 cn
c 1 c 2 c n 1/n .
n
In other words, the arithmetic mean of the numbers c 1 , c 2 , , c n is not less than the geometric mean.
We define
P x 0 , x 1 , , x n
to be the regular npartition of 0, 1 and we define f to be the step function on 0, 1 that takes
the constant value c j on each interval x j 1 , x j and we apply Part b. 285 Alternative 11 The RiemannStieltjes Integral
Some Exercises on Step Functions
1. True or false? If f is a step function on an interval a, b and c, d is a subinterval of a, b then f is a step
function on c, d . Solution: The statement is true.
c
d
b
a
Choose a partition P of a, b within which the function f steps and then refine P by adding to it the two
numbers c and d. Then drop all of the points of this partition that lie outside of the interval c, d and we
obtain a partition Q of c, d within which f steps.
2. True or false? If f is a step function then given any interval a, b , the function f is a step function on a, b . Hint: This statement is true. Write a short proof. Choose an interval c, d such that f is a step function on c, d and such that f x 0 whenever
x R c, d . Choose numbers p and q such that p is less than both of the numbers a and c and q
is greater than both of the numbers b and d p a c b d q Choose a partition P of the interval c, d such that f steps within P. If we add the two numbers p and
q to the partition P then we obtain a partition of the larger inte...
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 Fall '08
 STAFF
 Math, Calculus

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