  Holevo introduced a fidelity between quantum states that is symmetric and as
effective as the trace norm in evaluating their similarity. This fidelity is
bounded by a function of the trace norm, a relationship to which we will refer
as Holevo's inequality. More broadly, Holevo's fidelity is part of a
one-parameter family of symmetric Petz-R\'enyi relative entropies, which in
turn satisfy a Pinsker's-like inequality with respect to the trace norm.
Although Holevo's inequality is tight, Pinsker's inequality is loose for this
family. We show that the symmetric Petz-R\'enyi relative entropies satisfy a
tight inequality with respect to the trace norm, improving Pinsker's and
reproducing Holevo's as a specific case. Additionally, we show how this result
emerges from a symmetric Petz-R\'enyi uncertainty relation, a result that
encompasses several relations in quantum and stochastic thermodynamics.
