Bernstein type inequalities for derivatives of rational functions in spaces for

Abstract

It is shown that if is a rational function of degree , , , and , then for any (*)where and depends only on and . The problem of proving the inequality (*) was posed by Sevast'yanov in 1973. Up to the present, it has been solved for . In the case the inequality is not valid. Some applications of (*) to problems of rational approximation are also given in this paper. Similar questions are considered for the line and circle.