Estimation of an algebraic polynomial in a plane in terms of its real part on the unit circle |
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Authors: | A V Parfenenkov |
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Institution: | 1. Ural State University, pr. Lenina 51, Ekaterinburg, 620083, Russia
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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 Γr = z ∈ ?: ¦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 ||Pn||C(Γr) ≤ μ(r,n)||Re Pn||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. |
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