SHARP EXTENSIONS OF BERNSTEIN’S INEQUALITY TO RATIONAL SPACES Peter Borwein and Tam´ as Erd´ elyi Department of Mathematics and Statistics Simon Fraser University Burnaby, B.C. Canada V5A 1S6 Abstract. Sharp extensions of some classical polynomial inequalities of Bernstein are established for rational function spaces on the unit circle, on K := R (mod 2π), on [−1, 1] and on R. The key result is the establishment of the inequality |f ′ (z 0 )|≤ max           j=1 |a j |>1 |a j | 2 − 1 |a j − z 0 | 2 ,  j=1 |a j |<1 1 −|a j | 2 |a j − z 0 | 2          ‖f ‖ ∂D for every rational function f = pn/qn, where pn is a polynomial of degree at most n with complex coefficients and qn(z)= n  j=1 (z − a j ) with |a j |= 1 for each j , and for every z 0 ∈ ∂D, where ∂D := {z ∈ C : |z| =1}. The above inequality is sharp at every z 0 ∈ ∂D. 1. Introduction, Notation. We denote by P r n and P c n the sets of all algebraic polynomials of degree at most n with real or complex coefficients, respectively. The sets of all trigonometric polynomials of degree at most n with real or complex coefficients, respectively, are denoted by T r n and T c n . We will use the notation ‖f ‖ A = sup z∈A |f (z )| for continuous functions f defined on A. Let D := {z ∈ C : |z |≤ 1}, ∂D := {z ∈ C : |z | =1} Typeset by A M S-T E X 1