On monotone rational approximation

Let f be an absolutely continuous function on [0,1] satisfying f′∈Lp[0,1], p>1, Qn-be the set of all rational functions r=s/q, where s and q are polynomials of degree ⩽n. We prove: if f is a monotone function on [0,1], then there is a monotone rational function r∈Qn, such that∥f-r∥C[0,1]⩽c(p)n∥f′∥Lp[0,1],n=1,2,....