PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 3, Pages 879–887 S 0002-9939(02)06571-1 Article electronically published on June 12, 2002 BERNSTEIN–WALSH INEQUALITIES AND THE EXPONENTIAL CURVE IN C 2 DAN COMAN AND EVGENY A. POLETSKY (Communicated by Juha M. Heinonen) Abstract. It is shown that for the pluripolar set K = {(z,e z ): |z|≤ 1} in C 2 there is a global Bernstein–Walsh inequality: If P is a polynomial of degree n on C 2 and |P |≤ 1 on K, this inequality gives an upper bound for |P (z,w)| which grows like exp( 1 2 n 2 log n). The result is used to obtain sharp estimates for |P (z,e z )|. 1. Introduction If X is a non-pluripolar compact set in C k and P is a polynomial of degree n on C k , the Bernstein–Walsh inequality is (see [K]) |P (z )|≤‖P ‖ X e nVX(z) , (1) where ‖P ‖ X is the uniform norm of P on X and V X (z ) is the extremal function of X . For example, if z =(z 1 ,...,z k ) and X =Δ k = {z ∈ C k : |z j |≤ 1, 1 ≤ j ≤ k} is the unit polydisk, then V X (z )= L(z ) = max{log + |z 1 |,..., log + |z k |}. If X is pluripolar, then, in general, such estimates are impossible. For example, if X is any piece of an algebraic curve Γ = {(z,w) ∈ C 2 : P (z,w)=0}, where P is a polynomial, then ‖cP +1‖ X = 1 for every c> 0 and there are no upper bounds on cP + 1. We consider the case when Γ = {(z,w) ∈ C 2 : w = f (z )} and the compact set K = {(z,f (z )) ∈ C 2 : |z |≤ 1}, where f is an entire transcendental function. Then any non-trivial polynomial is not identically equal to 0 on K. Therefore a compactness argument shows that, for every n, there is a number c n > 0 such that for any polynomial P (z,w) of degree at most n the norm ‖P ‖ Δ 2 ≤ c n ‖P ‖ K . Hence for every (z,w) ∈ C 2 |P (z,w)|≤‖P ‖ K E n (f )e nL(z,w) , (2) where E n (f ) is the least value of c n . (See also Section 2.) Inequality (2) can be viewed as a transcendental global version of the Bernstein– Walsh inequality (1), provided that one can obtain good estimates for E n (f ). More- over, the numbers E n (f ) can serve as a measure of transcendency of f : A “less Received by the editors June 8, 2001 and, in revised form, October 18, 2001. 2000 Mathematics Subject Classification. Primary 41A17; Secondary 30D15, 30D20. The second author was partially supported by NSF Grant DMS-9804755. c 2002 American Mathematical Society 879 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use