Optimization of integral

From: Mark Flanagan (john_g_proakis_at_hotmail.com)
Date: 10/11/04 

Date: 11 Oct 2004 11:26:34 -0700

I am interested in proving the following conjecture. It seems like it
should have a "neat" proof, as opposed to, say, using calculus of
variations...

***********************************************************************

If G(x) is a given continuous function on x \in [0,1] satisying

\int_0^1 G(x) dx = 1

and

G(x) \ne 0 on x \in [0,1]

and Q(x) is allowed to be any continuous function on x \in [0,1]
satisying

\int_0^1 Q(x) dx = 1

Then,

\int_0^1 ( Q(x) / G(x) ) ^2 dx \ge 1

with equality iff Q(x) = G(x) for all x \in [0,1]

*************************************************************
Any ideas?

-- Mark

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