The degree of coconvex polynomial approximation

Let $f\in C[-1,1]$ change its convexity finitely many times in the interval, say $s$ times, at $Y_{s}:-1<y_{1}<\cdots <y_{s}<1$. We estimate the degree of approximation of $f$ by polynomials of degree $n$, which change convexity exactly at the points $Y_{s}$. We show that provided $n$ is sufficiently large, depending on the location of the points $Y_{s}$, the rate of approximation is estimated by the third Ditzian-Totik modulus of smoothness of $f$ multiplied by a constant $C(s)$, which depends only on $s$.