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 $$R^{\prime }(x)<\frac{9^n}{1-x^2}\Vert R\Vert ,x\in (-1,1).$$ The exponential dependence of constant factor on $n$ is shown, with sharp estimates for odd rational functions.

9 pages