🚀 go-pugleaf

I have a conjecture which I have been unable to find in the library, prove,
or discover a conterexample:

There exists some constant C > 0 such that for all non-negative functions
f(x) on [0,inf] which satisfy the inequality

INT{ f(x) x dx } <= C INT{ f(x) dx } ,

the inequality

INT{ f(x) dx }^2 <= INT{ f(x)^2 dx }

also holds.  All integrals are over [0,inf] and converge.

Physically, this means that for any positive distribution of mass f(x), if
the center of mass is less than C, then the mass is bounded by the L_2 norm.
The conclusion is similar to Cauchy-Schwarz, but with an interval of
integration longer than 1.

Actually, the problem I need to solve is much more complicated than this,
but I am hoping that I can learn something from the method of proof for
this, since it is very similar.

Brian Sutin				sutin@helios.ucsc.edu
Lick Observatory, UCSC			Santa Cruz, CA 95064
    Fred:   "May I rescue you?"
    Ginger: "No, thank you.  I prefer being in distress."