  For any 3-monotone on $[a,b]$ function $f$ (its third divided differences are
nonnegative for all choices of four distinct points, or equivalently, $f$ has a
convex derivative on $(a,b)$) we construct a cubic 3-monotone (like $f$) spline
$s$ with $n\in \Bbb N$ "almost" equidistant knots $a_j$ such that $$ \left\Vert
f-s \right\Vert_{[a_j,a_{j-1}]} \le c\, \omega_4
\left(f,(b-a)/n,[a_{j+4},a_{j-5}]\cap [a,b]\right), \quad j=1,...,n, $$ where
$c$ is an absolute constant, $\omega_4 \left(f,t,[\cdot,\cdot]\right)$ is the
$4$-th modulus of smoothness of $f$, and $||\cdot ||_{[\cdot,\cdot]}$ is the
max-norm.
