  We introduce new shape-constrained classes of distribution functions on R,
the bi-$s^*$-concave classes. In parallel to results of D\"umbgen, Kolesnyk,
and Wilke (2017) for what they called the class of bi-log-concave distribution
functions, we show that every $s$-concave density $f$ has a bi-$s^*$-concave
distribution function $F$ for $s^*\leq s/(s+1)$. Confidence bands building on
existing nonparametric bands, but accounting for the shape constraint of
bi-$s^*$-concavity, are also considered. The new bands extend those developed
by D\"umbgen et al. (2017) for the constraint of bi-log-concavity. We also make
connections between bi-$s^*$-concavity and finiteness of the Cs\"org\H{o} -
R\'ev\'esz constant of $F$ which plays an important role in the theory of
quantile processes.
