1873_solutions

That f c 0 using the fact that f is continuous at c

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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 n-partition 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 Riemann-Stieltjes 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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