# Integral Inequality

• December 14th 2006, 05:25 PM
ThePerfectHacker
Integral Inequality
I am trying to prove that the shortest distance between two points is a straight line. And it is almost complete, if I can use the following theorem.
Assume,
$f(x),g(x)$ are Integratable on $[a,b]$.
And,
$\int_a^b [f(x)]^2 dx\leq \int_a^b [g(x)]^2 dx$
Then is it true that,
$\int_a^b |f(x)|dx\leq \int_a^b |g(x)|dx$?

And one more,
$0\leq \int_a^b f(x)dx\leq \int_a^b g(x)dx$
Und,
$0\leq \int_a^b a(x)dx\leq \int_a^b b(x)dx$
Then,
$0\leq \int_a^b f(x)a(x)dx\leq \int_a^b g(x)b(x)dx$

I tried the Cauchy-Swarthz inequality because this is an inner product space but it does not help sufficiently.
• December 15th 2006, 04:45 AM
galactus
Hey PH. Here's a nice proof if you wanna check it out. Something you may like.

Proof shortest distance between two point is a straight line
• December 15th 2006, 08:49 AM
ThePerfectHacker
Quote:

Originally Posted by galactus
Hey PH. Here's a nice proof if you wanna check it out. Something you may like.

Proof shortest distance between two point is a straight line

Well, I did it with my professor. The proof is so much nicer. This happens to be a Calculus of Variations problem. He told me the standard way is to define $y$ "what you think is the answer is" in this case the line passing through the two points. And let $y+\delta y$ be any other path taken, where $\delta y$ is some smooth function. And then show the arc length of $y+\delta y$ is more than the arc length of $y$. I was able to do it by the use of the inequalities above. And if someone says they are valid (and I think they are) I can post my solution.