Necessary conditions for metrics in integral Bernstein-type inequalities

Let T(n) be the set of all trigonometric polynomials of degree at most n. Denote by Phi(+) the class of all functions phi: (0, infinity) -> R of the form phi(u) = psi(In u), where psi is nondecreasing and convex on (-infinity, infinity). In 1979, Arestov extended the classical Bernstein inequality parallel to T(n)(1)parallel to C <= n parallel to T(n)parallel to C, T(n) is an element of T(n), to metrics defined by phi is an element of Phi(+): integral(2 pi)(0) phi(vertical bar T(n)(1)(t)vertical bar)dt <= integral(2 pi)(0) phi(n vertical bar T(n)(t)vertical bar)dt, T(n) is an element of T(n). We study the question whether it is possible to extend the class Phi(+), and prove that under certain assumptions Phi(+) is the largest possible class. (C) 2010 Elsevier Inc. All rights reserved.