Bernstein type inequality in monotone rational approximation

TL;DR

This paper establishes a Bernstein-type inequality for monotone rational functions, showing an exponential dependence of the constant factor on the degree, with sharp estimates for odd functions, advancing understanding of rational approximation.

Contribution

It provides a new Bernstein inequality for monotone rational functions with explicit exponential bounds and sharp estimates for odd functions.

Findings

The inequality holds for increasing rational functions of degree n.

The constant factor depends exponentially on n.

Sharp estimates are obtained specifically for odd rational functions.

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.