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.