  We investigate bounds in the transmission of classical information through
quantum systems. Our focus lies in the generalized Holevo theorem, which
provides a single-letter Holevo-like inequality from arbitrary quantum distance
measures. Through the introduction of the $\alpha$-$z$-R\'enyi relative
entropies, which comprise known relevant quantities such as the R\'enyi
relative entropy and the sandwiched R\'enyi relative entropy, we establish the
Holevo-R\'enyi inequality. This result leads to a quantum bound for the
$\alpha$-mutual information, suggesting new insights into communication channel
performance and the fundamental limits for reliability functions in memoryless
multi-letter communication channels.
