Sharp Variance-Entropy Comparison for Nonnegative Gaussian Quadratic Forms

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