  We consider the maximum entropy problems associated with R\'enyi $Q$-entropy,
subject to two kinds of constraints on expected values. The constraints
considered are a constraint on the standard expectation, and a constraint on
the generalized expectation as encountered in nonextensive statistics. The
optimum maximum entropy probability distributions, which can exhibit a
power-law behaviour, are derived and characterized. The R\'enyi entropy of the
optimum distributions can be viewed as a function of the constraint. This
defines two families of entropy functionals in the space of possible expected
values. General properties of these functionals, including nonnegativity,
minimum, convexity, are documented. Their relationships as well as numerical
aspects are also discussed. Finally, we work out some specific cases for the
reference measure $Q(x)$ and recover in a limit case some well-known entropies.
