Bi-$s^*$-concave distributions

We introduce a new shape-constrained class of distribution functions on R, the bi-$s^*$-concave class. In parallel to results of Dumbgen, 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 $F$ and that every bi-$s^*$-concave distribution function satisfies $gamma (F) le 1/(1+s)$ where finiteness of $$ gamma (F) equiv sup_{x} F(x) (1-F(x)) frac{| f (x)|}{f^2 (x)}, $$ the CsorgH{o} - Revesz constant of F, plays an important role in the theory of quantile processes on $R$.