Bernstein type inequality in monotone rational approximation

Abstract: 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.