I'm seeking the references concerning on the regularity analysis of concave envelopes, *i.e.* given some measurable function $f:\mathbb R^d\to\mathbb R$ that is bounded from above ($d=1$ or $d\ge 1$), denote by $f_{\#}$ its concave envelope, *i.e.*

- $f_{\#}:\mathbb R^d\to\mathbb R$ is concave and $f_{\#}\ge f$;
- For any concave function $g:\mathbb R^d\to\mathbb R$ s.t. $g\ge f$, one has $g\ge f_{\#}$.

Assume that $d=1$. I'm mainly interested in the boundedness of the (left/right) derivative $f_{\#}'$. Under what kind of conditions on $f$, we can show that $f_{\#}'$ is uniformly bounded? Any related reference is welcome! Thanks a lot!