Three Problems on Integration -- Rudin

[Douglas Squirrel]

I'm working my way through Rudin's *Principles of Mathematical Analysis* this
summer. I've gotten partway through each of the following three problems
from Chapter 6, but can't seem to finish. Can you offer any hints? No
solutions, please; I'd like to solve the problems myself. If you feel you
*must* post a solution, please label it as such so I can ignore it.

Thanks!

(10) Let p and q be positive real numbers such that (1/p) + (1/q) = 1.
I've been able to show
(a) for any u, v >=0, uv <= u^p/p + v^q/q. Equality holds
iff u^p = v^q.
and
(b) Given functions f,g >=0, both integrable wrt a function a, with
\int_a^b f^p da = \int_a^b g^q da = 1.
Then
\int_a^b fg da <= 1.
But I can't solve
(c) Given complex functions f and g, both integrable wrt a function a.
Then
|\int_a^b fg da| <= (\int_a^b |f|^p da)^{1/p} \cdot (\int_a^b |g|^q da)^{1/q}.

(Note that "|" is the absolute value symbol, and \cdot is TeXspeak for a dot--
i.e. the mathematician's "times" sign.)

(13) Let f(x) = \int_x^{x+1} sin (t^2) dt. I've been able to get (a), which
says that |f(x)|<1/x for all x>0, and (b), which says that
2 x f(x) = cos(x^2) - cos[(x+1)^2] + r(x)
where |r(x)|<c/x. However, I can't find the upper and lower limits of xf(x)
as x->oo, or determine whether \int_0^{oo} sin(t^2) dt converges. (Note that
oo is infinity, if it doesn't look like it on your screen :)

(15) Given f real and differentiable, f(a) = f(b) =0, f' continuous. Suppose
\int_a^b f^2(x) dx = 1. (Here f^2(x) means f(x) squared.) I've been able to
show that
\int_a^b x f(x) f'(x) dx = -(1/2)
but not that
\int_a^b [f'(x)]^2 \cdot \int_a^b x^2 f^2(x) dx > 1/4.

I can see that (10) and (15) are related, but not how. Also, can anybody
explain the point, if any, of (15)? Or is it just a "plug and chug" problem?

Thanks very much!

--dsqu...@reed.edu

[SunsetGun]

In article <33ek42$6...@scratchy.reed.edu>, dsqu...@reed.edu (Douglas
Squirrel) writes:

I'm working my way through Rudin's *Principles of Mathematical Analysis*
this
summer. I've gotten partway through each of the following three problems
from Chapter 6, but can't seem to finish. Can you offer any hints?

Been a while but here goes:

OK on 10(c):

Hint #1: The result 10(c) is called Holder's Inequality. If you really get
stuck, you can probably find a proof in another text.

Hint #2: The cases where f or g are zero on a set of measure zero are
trivial.

Put F = |f| over (\int_a^b |f|^p da)^{1/p}

Put G = |g| over (\int_a^b |g|^q da)^{1/q}

Apply the result of 10(b) with F and G.

on 13:

I'll work on it. I'm a workin' man. Not much time for mathematical
amusements. My intuition is that the limsup is 1 and the liminf is -1, and
that the integral 0^oo does not converge. Pretty sure about the limits.
Less sure about the integral.

on 15(b):

Hint #1: -(1/2) = (\int_a^b xff') implies (1/2) <= (\int_a^b |xf||f'|)

Hint #2: F = |xf|, G = |f'|, apply Holder with p = q = 2. This is the
connection with problem #10, which forms an outline of a proof of Holder's
Inequality. This problem (15), part (b) at least, it would appear, is just
a concrete example of using Holder (for my money, ya never can get enough
of that sort of thing).

Working through Rudin, eh? Admirable.

Take Care,

Chris (SunsetGun)