Sharp Variance-Entropy Comparison for Nonnegative Gaussian Quadratic Forms

In this article we study weighted sums of
$n$
i.i.d. Gamma(
$\alpha $
) random variables with nonnegative weights. We show that for
$n \geq 1/\alpha $
the sum with equal coefficients maximizes differential entropy when variance is fixed. As a consequence, we prove that among nonnegative quadratic forms in
$n$
independent standard Gaussian random variables, a diagonal form with equal coefficients maximizes differential entropy, under a fixed variance. This provides a sharp lower bound for the relative entropy between a nonnegative quadratic form and a Gaussian random variable. Bounds on capacities of transmission channels subjects to
$n$
independent additive gamma noises are also derived.