Re: Optimization of integral

From: G. A. Edgar (edgar_at_math.ohio-state.edu.invalid)
Date: 10/11/04 

Date: Mon, 11 Oct 2004 14:58:52 -0400

In article <198e254c.0410111026.3d8052f6@posting.google.com>, Mark
Flanagan <john_g_proakis@hotmail.com> wrote:

> 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

counterexample...

G(x) = x+(1/2); Q(x) = (7/4)x+(1/8); 

-- 
G. A. Edgar                               http://www.math.ohio-state.edu/~edgar/