Univariate right fractional polynomial high order monotone approximation

Let f ∈ Cr([−1,1]), r ≥ 0 and let L* be a linear right fractional differential operator such that L*(f) ≥ 0 throughout [−1,0]. We can find a sequence of polynomials Qn of degree ≤ n such that L*(Qn) ≥ 0 over [−1,0], furthermore f is approximated right fractionally and simultaneously by Qn on [−1,1]. The degree of these restricted approximations is given via inequalities using a higher order modulus of smoothness for f(r)