  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.
