Bernstein type inequality in monotone rational approximation

The following analog of Bernstein inequality for monotone rational functions is established: if $R$ is an increasing on $[-1,1]$ rational function of degree $n$, then $$R'(x)<\frac{9^n}{1-x^2}\|R\|,\quad x\in (-1,1).$$ The exponential dependence of constant factor on $n$ is shown, with sharp estimates for odd rational functions.