Estimation of an algebraic polynomial in a plane in terms of its real part on the unit circle

Abstract

We consider the class P _n ^* of algebraic polynomials of a complex variable with complex coefficients of degree at most n with real constant terms. In this class we estimate the uniform norm of a polynomial P _n ∈ P _n ^* on the circle Γ_r = z ∈ ℂ: ¦z¦ = r of radius r = 1 in terms of the norm of its real part on the unit circle Γ₁ More precisely, we study the best constant μ(r, n) in the inequality ||P_n||C(Γ_r) ≤ μ(r,n)||Re P_n||C(Γ₁). We prove that μ(r,n) = rⁿ for rⁿ⁺² − r ⁿ − 3r² − 4r + 1 ≥ 0. In order to justify this result, we obtain the corresponding quadrature formula. We give an example which shows that the strict inequality μ(r, n) = r ⁿ is valid for r sufficiently close to 1.