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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: [mg67361] Re: [mg67352] Help: ratio of integral of f(x)^2 to square of integral of f(x)
  • From: "Carl K. Woll" <carlw at wolfram.com>
  • Date: Tue, 20 Jun 2006 02:14:42 -0400 (EDT)
  • References: <200606190401.AAA21438@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

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.

You are trying to solve a variational problem, and the package 
Calculus`VariationalMethods` may be helpful.

However, the ratio is clearly not always greater than 1. Suppose f[x] 
yields a ratio of r>1. Then g[x]=f[x]/r^2 will yield a ratio smaller than 1.

Carl Woll
Wolfram Research


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