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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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