# Prove that a flat shape minimizes a functional

The following functional arises in an information theoretic problem that I work on currently.

$$I(G(\omega)) = \int_{-\kappa\pi}^{\kappa\pi} \frac{A}{G(\omega)+A}d\omega-\frac{\left| \int_{-\kappa\pi}^{\kappa\pi} \frac{A}{G(\omega)+A}\exp(-i\omega)d\omega\right|^2}{ \int_{-\kappa\pi}^{\kappa\pi} \frac{A}{G(\omega)+A}d\omega},$$

where $\kappa<1$, $A>0$, and $G(\omega)\geq 0$.

Now I would like to minimize $I(G(\omega))$ under the constraint of unit area of $G(\omega)$, i.e., $$\int_{-\kappa \pi}^{\kappa \pi} G(\omega)d\omega=1.$$

My hypothesis is that a flat $G(\omega)=1/2\kappa\pi$ is optimal, but I cannot prove that (Matlab hints towards it).

I can take the functional derivative and then apply the Lagrange multiplier method. But then I get stuck. A flat shape appears to be a stationary point (the functional derivative is zero), but how to prove global optimality? Is $I$ perhaps convex...?

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Can you please give here the Euler-Lagrange equation you obtained? – timur Aug 14 '12 at 1:16
Crossposted from physics.stackexchange.com/q/33632/2451 – Qmechanic Sep 15 '14 at 0:23