  Information theory has become an increasingly important research field to
better understand quantum mechanics. Noteworthy, it covers both foundational
and applied perspectives, also offering a common technical language to study a
variety of research areas. Remarkably, one of the key information-theoretic
quantities is given by the relative entropy, which quantifies how difficult is
to tell apart two probability distributions, or even two quantum states. Such a
quantity rests at the core of fields like metrology, quantum thermodynamics,
quantum communication and quantum information. Given this broadness of
applications, it is desirable to understand how this quantity changes under a
quantum process. By considering a general unitary channel, we establish a bound
on the generalized relative entropies (R\'{e}nyi and Tsallis) between the
output and the input of the channel. As an application of our bounds, we derive
a family of quantum speed limits based on relative entropies. Possible
connections between this family with thermodynamics, quantum coherence,
asymmetry and single-shot information theory are briefly discussed.
