  In this paper, we introduce new Stein identities for gamma target
distribution as well as a new non-linear channel specifically designed for
gamma inputs. From these two ingredients, we derive an explicit and simple
formula for the derivative of the input-output mutual information of this
non-linear channel with respect to the channel quality parameter. This relation
is reminiscent of the well-known link between the derivative of the
input-output mutual information of additive Gaussian noise channel with respect
to the signal-to-noise ratio and the minimum mean-square error. The proof
relies on a rescaled version of De Bruijn identity for gamma target
distribution together with a stochastic representation for the gamma specific
Fisher information. Finally, we are able to derive precise bounds and
asymptotics for the input-output mutual information of the non-linear channel
with gamma inputs.
