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

*To*: mathgroup at smc.vnet.net*Subject*: [mg67360] Re: Help: ratio of integral of f(x)^2 to square of integral of f(x)*From*: dh <dh at metrohm.ch>*Date*: Tue, 20 Jun 2006 02:14:35 -0400 (EDT)*References*: <e7589k$l5d$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Hi Ronnen, this is certainly wrong. Consider b-a small, say dx, then we can approximate the integral from 0 to dx by: f[0] dx: (1) (b-a) Integral[ f(x)^2 dx, {x, a, b} ] -> dx^2 f^2[0] (2) Integral[ f(x) dx, {x, a, b} ] -> dx f[0] Now, if dx is small enough, the quadratic term is smaller than the linear one. Here is an example: for f[x_] = Sin[x] the ratio is <1 for a< approx 1.4 and >1 otherwise. Daniel 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. >