Metric entropy of the class of probability distribution functions on $[0,1]$ with a $k$-monotone density is studied through its connection with the small ball probability of $k$-times integrated Brownian motions.
"Entropy Estimate for $k$-Monotone Functions via Small Ball Probability of Integrated Brownian Motions." Electron. Commun. Probab. 13 121 - 130, 2008. https://doi.org/10.1214/ECP.v13-1357