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.