  Quantum generalizations of Renyi's entropies are a useful tool to describe a
variety of operational tasks in quantum information processing. Two families of
such generalizations turn out to be particularly useful: the Petz quantum Renyi
divergence $\bar{D}_{\alpha}$ and the minimal quantum Renyi divergence
$\tilde{D}_{\alpha}$. In this paper, we prove a reverse Araki-Lieb-Thirring
inequality that implies a new relation between these two families of
divergences, namely that $\alpha \bar{D}_{\alpha}(\rho \| \sigma) \leq
\tilde{D}_{\alpha}(\rho \| \sigma)$ for $\alpha \in [0,1]$ and where $\rho$ and
$\sigma$ are density operators. This bound suggests defining a "pretty good
fidelity", whose relation to the usual fidelity implies the known relations
between the optimal and pretty good measurement as well as the optimal and
pretty good singlet fraction. We also find a new necessary and sufficient
condition for optimality of the pretty good measurement and singlet fraction.
