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.