Necessary conditions for metrics in integral Bernstein-type inequalities

Let TnTn be the set of all trigonometric polynomials of degree at most nn. Denote by Φ+Φ+ the class of all functions φ:(0,∞)→Rφ:(0,∞)→R of the form φ(u)=ψ(lnu)φ(u)=ψ(lnu), where ψψ is nondecreasing and convex on (−∞,∞)(−∞,∞). In 1979, Arestov extended the classical Bernstein inequality ‖Tn′‖C≤n‖Tn‖C, Tn∈TnTn∈Tn, to metrics defined by φ∈Φ+φ∈Φ+: ∫02πφ(|Tn′(t)|)dt≤∫02πφ(n|Tn(t)|)dt,Tn∈Tn. We study the question whether it is possible to extend the class Φ+Φ+, and prove that under certain assumptions Φ+Φ+ is the largest possible class.