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MathGroup Archive 2006

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Re: Help: ratio of integral of f(x)^2 to square of integral of f(x)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67362] Re: Help: ratio of integral of f(x)^2 to square of integral of f(x)
  • From: "jbaker75 at gmail.com" <jbaker75 at gmail.com>
  • Date: Tue, 20 Jun 2006 02:14:43 -0400 (EDT)
  • References: <e7589k$l5d$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Hi Ronnen -

I do not believe it is true for any function f(x). Consider f(x) = 1/2
on the interval [0,1]. Then:

Integral[ f(x)^2 dx, {x, a, b} ] = 1/4
Integral[ f(x) dx, {x, a, b} ] = 1/2
(b-a) = 1

so that r = 1/2.

Sincerely,
Jeff Baker

ronnen.levinson at gmail.com wrote:
> Hi.
>
> I'm trying to determine whether the following ratio
>
> r = (b-a) Integral[ f(x)^2 dx, {x, a, b} ] /
>               Integral[ f(x) dx, {x, a, b} ]
>
> is always greater than or equal to one for 0 < f(x) <= 1. All values
> all real.
>
> I've obtained r>=1 for all tested choices of f(x), but seek guidance to
> find the general answer.
>
> Yours truly,
>
> Ronnen Levinson.
> 
> P.S. E-mailed CC:s of posted replies appreciated.


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