In the literature, is there any asymmetric $S$-shaped function that maps the interval $[0, 1]$ to interval $[0, 1]$?
Unfortunately I can't post figure so I just describe what I mean in text. The function I want should be monotonic increasing. In the meanwhile, for a real number $c < 0.5$, the function
$$0 \mapsto 0, \quad c \mapsto 0.5, \quad 1 \mapsto 1.$$
Moreover, the function is smooth on its domain, concave on interval $[0, c]$ and convex on interval $[c, 1]$.
By the way, can it be seen as a sort of link-function?