  We present a new approach to response around arbitrary out-of-equilibrium
states in the form of a fluctuation-response inequality (FRI). We study the
response of an observable to a perturbation of the underlying stochastic
dynamics. We find that magnitude of the response is bounded from above by the
fluctuations of the observable in the unperturbed system and the
Kullback-Leibler divergence between the probability densities describing the
perturbed and unperturbed system. This establishes a connection between linear
response and concepts of information theory. We show that in many physical
situations, the relative entropy may be expressed in terms of physical
observables. As a direct consequence of this FRI, we show that for steady state
particle transport, the differential mobility is bounded by the diffusivity.
For a virtual perturbation proportional to the local mean velocity, we recover
the thermodynamic uncertainty relation (TUR) for steady state transport
processes. Finally, we use the FRI to derive a generalization of the
uncertainty relation to arbitrary dynamics, which involves higher-order
cumulants of the observable. We provide an explicit example, in which the TUR
is violated but its generalization is satisfied with equality.
