SHARP EXTENSIONS OF BERNSTEINS INEQUALITY TO RATIONAL SPACES

Sharp extensions of some classical polynomial inequalities of Bernstei n are established for rational function spaces on the unit circle, on K = R (mod 2 pi), on [-1, 1] and on R. The key result is the establish ment of the inequality [GRAPHICS] for every rational function f = p(n) /q(n), where p(n) is a polynomial of degree at most n with complex coe fficients and [GRAPHICS] with \a(j)\ not equal 1 for each j, and for e very z(0) is an element of partial derivative D, where partial derivat ive D = {z is an element of C:\z\ = 1}. The above inequality is sharp at every z(0) is an element of partial derivative D.